Transcript for:
Calculus Basics: Limits & Derivatives

Today in this video we will tell you about your limits and We are going to do derivatives of class 11th This is one of the most important chapters why because from here your calculus it starts i know right now you You will understand things at 20, so right now I said keep patience at the end of all of it There won't be a single doubt left that you are here x that you can keep the values that are in its domain if it happens then if you put two in place of x then What will be the value of the function h friends If h = 0 is not written then you would could not do this, he said that The value is going from zero to near zero Exactly is not zero so here I am I am cutting it, this is a very important matter There are small details in maths How are you everybody? So friend, today we are going to Your Limits and Derivatives in Video I am going to do a lot of work in this chapter There was a demand but anyway I will just give you this I will say that this is one of the most for class 11th Why are these chapters important This is where your call loss starts Hey friend, you must have heard about the differentiators of all of them friends it starts from here okay Everyone's foundation is not the limits here I am okay why do I read limits Why study what are derivatives What does Limits mean in Derivatives means today in this lecture you will I will teach you not to do anything like this learn it okay my promise is that as many There are also concepts that you might have to memorize. Let's go, I will try my best to ensure that boss You should understand, you should feel that brother If you are solving the question, then you You need to know what's going on bro ok just one thing in the question I will ask that brother look this is the promise that this You don't have to do anything else after the lecture. Limits Derivatives Zhe after Lecture It may seem very easy but your patience this lecture is going to be a bit long It is okay to have patience, patience will take care of everything Things A to Z Basic to Advanced all clear I will do it in this one short lecture Start, do you have to sit with a pen and paper? Come on boss, let's start first friend brother i have to read only two topics in the chapter Limits and derivatives are the most First, we will study the limits for one and a half hour, right? you will go within your limits hey where is this pen The pen is fine, now listen to me carefully as I am saying, whatever I am saying just keep listening to that okay first of all i Always remember whose limit I remove I will find the limit of the function friends Whose limit do I find for the function listen i know you are 20 things right now You will understand, that's when I told you to have patience At the end of all of it there won't be a single doubt It will go well, so in such a wonderful way Boss, I will teach you to teach someone else also Can explain limits and derivatives it's better than any teacher of yours, okay I am Find the limit of any function you guys know the function remember the second Chapter Relation Function What is a Function okay where do we find the limit at a point where do we find the limit at a I will tell you the point, I will tell you everything, okay that's a lot The important question is why are we getting wiped out remove it because listening to function text we limit at that point we limit at that point but we remove it because the function tex because function tex indeterminate what did i say form function text indeterminate form at that point in determinate form at that point very important thing because the function is that At the particular point where your limit We are taking out that point, indeterminate what does it mean it takes form I will explain it to you a little late in the chapter. how is this how do you remove the chapter how to remove the limit now pay attention to what i am saying Let's start from the very basics Friends, I had some function as if fxxx.pro what will mean the value of the function at two What will be the value of the function two on two now what is the domain of a function what is the domain I used to see you coming to some function I want it but okay let's not go to hell with the domain know so simple domain means the values of x which Here you can put the values of x its domain is okay so instead of x if you keep two then what is the value of the function It will come, hey, if you simply put two in place of x The value of the function 2+ 5 27 is sen is ok but i am not interested in that The value of the function at t or at any other point But what will happen? What did I say about this? I am not even interested in the function At value u or at some Dimka point 5/2 - 100 1 lakh 1 crore 3 crore 5/2 -5 What is the value of the function at any point It will come, I am not interested in that I am interested to know that The value of the function if the value of x is close to two if I let me talk about two point the value of x when two If the value of the function is around I am interested in understanding this Come, now you have to listen carefully, everything is explained I'll pretend I made a number line now listen i wrote zero here one was written here two was written here Friends, I am interested around two What will be the value of the function around just keep listening okay now I'm going to tell you two How can I approach friends? I am too sorry I am too left I can approach from the right hand side or So I'm going to go from the left hand side towards two. I can yes or no, are you understanding brother I am I'm interested in the thing when I when I If I go from the left hand side to two then Towards which value will the function tend when I turn to the right hand side what is the value of the function if I move towards I am moving towards value in this matter I am interested why am I interested I will tell you the reason for this, is it correct, do you understand this? Have you come, now take this as an example I said two to two on the right hand side right Come on, let me show you from here to there hmm here it is for example For example, I took a function I use fxxx.pro from two sides friends I can approach from either the right hand side from I can move to three either From the left hand side I move towards three I can listen again carefully what did you say this is your number line is ok I want to hear your cry, your forest, yours two or your three or your four either I Can move to three from the right hand side I'm either on the left hand side I'm on three Now let's say if I move left I'm moving from the hand side to three. What is happening on the left hand side? what do you mean first of all i put 2.9 in place of x kept So the value of the function 2.9 + 3 will be 5.9 then 2.9 to 2.9 just a slightly larger value If we kept it 2.99 then we kept the value of x slightly higher fx.com 2.9 it will be something like 2.99 it will be something like 2.99 then this 2.99 so as you go left hand If you're going from side to side three, then the function Towards which value is the value of tell me as you go from the left hand side I am going towards three, are you seeing this? the value of x is increasing the value of x is increasing where are you going going towards three x The value of this will go towards your three friends. then what is the value of the function I am going that way, just tell me about the function Is the value going towards your six I understand this and I want to say this this is what is called limit this value They say limit and how do they write like this limit x tends to pay attention to x x key value three but not x tends to x Going left towards approachable three Three minuses from the hand side mean from the left hand side then the value of the function what is the limiting value of the function Friends 6 did you understand this? Did you understand this, brother, you are divided If you write thr here then the function key The value 3 will give 6 but this means what does this whole thing mean Should one know how to read and understand that when the boss The value of x is calculated from the left hand side when I change the value of x Take it from the left hand side towards three so the value of the function is going towards six is going this is what is called a function This is called the limiting value of a function Now I understand the limiting value of But look what is happening brother either I Approach three from the left hand side I will either do a three from the right hand side now I'll approach the right hand side how can i do it from 3.1 then 3.01 then 3.00 Do you understand brother what is happening? Look, you are going towards three only. No, first it was written 3.1, then 3.01, then 3.00 wrote then wrote 3.00 then wrote 3.00 then 3.00 is written so what am I doing here Here I am going from right hand side to right hand side I am going from the side towards thi, do you understand? then where is the value of the function going tell me brother if you want to keep 3.1 then what is the function The value of the function will be 6.1 if you substitute 3.01 The value will be 6.01 If you substitute 3.1 then the value of the function will be 6.0001 will come If you write 3.000 then the value of the function will be 6.00 Please tell me where the value of the function will come from it is going if you take three from the right hand side then the value of the function Here also two is going towards 6, how come I understood it just now, earlier it was 6.1 then It became smaller, then smaller, then smaller like this while doing this it is going towards six so this I will write it, do I write it like this what that Boss what is the limit x tends to 3 positive This means when I am on the right hand side I am going towards thr meaning of positive Going to three from the right hand side Negative means left hand side side going towards three positive negative means this There is no actual positive or negative here pe negative means i am on left hand side I'm going from to three and here I am Positive means I am from the right hand side I'm going towards three, if you understand then the function what is the value of the function Si is coming and this is what I call function Limiting value of Limiting value at 3 What did you say about the limiting value of the function t3 what did i say i used limiting function value at 3 now listen carefully Right now I am going to talk about a particular sect. Right now I am at a particular point I am talking about it, pay attention to what happened when you are talking about this function if you are talking about this function then So going from the left hand side towards three Even if it is there, the value of six is coming in the right hand Even if you are going from side to side three The value six is coming in both the cases The value is coming same, in this case you say yes or limit exists what do you say does your limit exist at 3 you will say limit x tends to 3 now you What does it mean if positive is not written as negative? Let me tell you that the value of the function is coming out to be 6 This means limits exist, did you understand? okay remember this and what is the important thing Friends, this is called left hand limit. Friends, this is called left hand limit. Left hand limit means left hand side we are moving from to th this is called Friends what does the right hand limit say Right Hand Limit in Short Form R A L Always remember one very important thing what if if note down the left hand limit is equals to right hand limit is equals to find a fine value is equal to any fine it If the value is like this then what do I say That Give and Take Limit Everything will be clear, just keep listening The reason why I am taking this out Let me explain, I say that limits exist okay patience is the key i know You must be having doubts about the limit If you want to take it out, I will tell you everything, don't worry tell you everything i will tell you but first this Focus on the board, what did I read bro i am not interested in that one What is the value of the function at a particular point it is coming i am interested in this That I took a three at that point From the right hand side when I go to three Then the value of the function will tend towards which value? When I turn three from the left hand side if I go in that direction then what is the value of the function I'm heading towards this thing I am interested and if from both sides from left hand side to right hand side The same value is coming from both sides of the function that's what I say and that value is finalized There should be value, so I say that limits exist limits exist There is a limit on that path but limit exists on that path and What will be the value of limit, it will be six because the function is going towards six it's ok from both sides right hand limit You understood the left hand limit, okay, let's see just keep watching okay now what does it mean did you understand it carefully once again since I have already written it to you pay attention what does it mean friends relax First understand what it means that boss that boss when the value of x is left hand is going from side a to side a when the x The value goes from the left hand side towards a then the value of the function is going towards m when the value of x is from the right hand side is going towards a then the value of the function is n is heading towards and this is what is called The value of limit is n okay here I have Look, I even wrote it from the left hand side, x If the value of is going towards a then the function The value of is going towards m when right The value of x from the hand side is going towards a then the value of the function is going towards n I understood it, it became clear, okay hey boss okay very good okay now Why do you set limits? Just think about it. I understand okay what happens what happens that the function does not function at that particular point gives you the indeterminate form now What is indeterminate form for a while I will tell you but understand that I am from a particular Limiting the value of a function at a point Why do I want to know the value of something? As you can imagine friends, consider this as your road there is a tunnel there is a tunnel tunnel ok this is your road Tunnel road is ok now listen friends here But there are cameras installed everywhere, here you There are cameras like these everywhere, keep listening Ok, you just need to understand the concept here. The question is very easy, it is okay to do it There are cameras installed everywhere, now you guys are making me think You guys should think about it and then tell me, ok It seems like an accident happened here friends Accident happened at a particular point, ok But if an accident happens, what do you do first? Will you do it brother, you want to know this, you are the police Are you in the police or are you investigating You are an officer, you want to know that brother Obviously how did the accident happen I understand well how the accident happened So what is the first thing you check? tell me you were not present there then you tell me what will you check, will you check the cameras will you do it Obviously you will check the cameras and which cameras which hits at this point It seems that the camera at this point was accidentally damaged I don't know if it came out or if it didn't take any footage If you found it at that time then what would you do, a simple This is common sense so you are on its side You will check the side cameras would you try to find out from this I will repeat what happened at the point You can use the side camera as if it were this camera From You will try to find out whether this what is happening at a particular point because at this particular point your camera If it is closed then you cannot find it at that point. If you can find out exactly what happened then you are around are you trying to find out from It is happening okay friends, limits are also same is basically that at a particular point The function cannot return its value, you do this Imagine if we can find out the value around it try to do that around him Understand what value the function is giving you I'm coming in what am I saying okay Now you tell me this left hand limit right I tell you the meaning of hand limit as if Have you checked this camera, did you see that brother A yellow car is going very fast from here it's coming out okay it's going very fast okay And then if you check this camera then From here also you will see a yellow colour car it's going out very fast so if you From both sides, from left hand side to right If the same thing is visible from the hand side then Do you accept this conclusion? Maybe yes for responsible accident or Neither yes nor no, am I right or not this is what is called that limits exist listen carefully if the left hand Limit right hand limit both are same She is telling that both are giving the same value So you will say that limit exists but you Imagine that brother, yellow color comes from here that the car came out very fast, okay but I don't know There was no red car here, but When this camera was seen, the red coloured car If it is coming out then brother you are confused now Whether you went or not, it definitely happened, this means which is visible from the left hand side and What is visible from the right hand side The left hand limit is not the same as the right hand limit If the limit is not the same then you will say limit it does not exist because you already know Boss, you are not able to figure it out, you don't even know I can guess that exactly at that point what happened do you understand what i said If I am here then I have to exist within limits I don't want to exist, I hope you understand. It will be clear now in reality What does this indeterminate form mean? learn more about what it is Okay friend, let's go now, let's do one thing first what are these determinant forms why is the function not having a value at that point The function is able to give the value at that point Why can't you give me the company of those around me Why do we need to know the values I will tell you a little later, right now I don’t know how to remove limits and how to leave How to calculate right hand limit and because aya first ask some questions take it or else you will keep understanding You will say that it has been half an hour, I understand You are being understood, look comfortably Ok, now see how I find out hmm i saw a way boss I saw that boss, believe me, believe me I have been given a function, 3 has been given on it I want to find the limiting value of a function limiting means when from the left hand side The function when I from the left hand side If I am going towards value then what is the function giving value when from right hand side I'm going to go towards that value. what value is the function giving me Interested in knowing the value If it is there then boss I won't do this every time once in every question that boss function came at some point, at some point slowly I'm going from the left right hand side and Go slowly to the point from the left hand side I am here and I am seeing that brother is having a function What is the value coming and I am seeing that bro the value of the function is not coming I would do that, what would I do instead instead friends Just smaller than I asked at any point value and just put the big value in the function let me try it ok ok how to do it Just pay attention as if you got a function Limit ex tends to believe that you are in a particular I want to find out at point A that the boss What will be the value of the function, then what will you do You should do it from the left hand side to the same side Have to go and from the right hand side to the same side I need to go and find out what the function is What is limiting value and if the same If there is value from both the sides then the function Limits do exist, right? How do you know? what will you do first How do you calculate the left hand limit? ho you first take out the left hand limit how when you go from the left hand side to a If you are going then what is the value of the function So brother, it means from the left hand side Let's say yours is a from the left hand side means a cy means just a smaller number than a So now first of all listen here this is important what is it that here i am a small h I introduce you to someone whose value is very high It is small but positive whose value is very high It is very small but it is positive means it is greater than zero okay do you understand now tell me this is two as if it's a If ू is written in any place then just Which will be the smaller value 1.99 Do you understand what I am saying? ok so what do I do anyway Am A sorry limit x tends to a negative mean When I go from the left hand side towards A I am here, what are you doing What do you just do here? What is the value of f? it will be an attempt to remove it, okay you what will you do you do x in place of x you will put a - h in place of x I will put a - h in place of x ok a - h in place of x means a value slightly smaller than a A slightly smaller value means something here here i know the value of the function I'm trying to do it okay so keep it up just think what will be the limit here now Understand that the value of x is coming on one side of x If the value of h goes towards a then the value of h will go towards will go towards how nothing the value of x is a If you believe this, then you will write a in place of x What will be the value of h What will be the value of h If you go here then friends instead of x you have put h you introduced h instead of x If you introduce then instead of x you use h tends I will write 20, just keep listening slowly Look, you are reading something new for the first time. The problem is not thinking that the boss Hey friend what is going on hey I understand everything He will come, just keep waiting, have patience, trust me okay on pass i will say it again the value of x is a or is going towards a then x if I put a here then what is the value of h She will come and cry, see if you come here then If I went there then a - a 0 h 0 so limit h tends to 0 and instead of x you wrote a - h and Or find out, did you understand what I said I told you and or find out how to do it I will tell you about the left and right hand limits If I talk about it, then right hand limit means right When going in one direction from the hand side then just Put a value slightly greater than a What will be the value slightly greater than a in a small number plus two small numbers I call that lighter number h in a minus a small number, that small number To whom do I say h, did you understand? Yes it is yours. Remember this is positive, how to remove it limit x tends to a positive mean right from the hand side when I'm moving in the same direction as a So what is the value of the function I am getting? what do i do i replace x with a + h slightly with a brother now instead of x I need a So do not keep a number slightly greater than a If we want to put a then if the value of x is a if the value of h is 0 then you can change it doge limit h tends to 0 f of a + h you Now we will take it out, I will tell you how to take it out I speak in this way that in this manner you will evaluate your limits, okay do not remove it at a particular point because Indeterminate form will come, I will tell you what it is I am trying to find out the surrounding values. How do I find out from that point? A slightly smaller number, slightly above that point I kept a big number and saw if it was the same at both places Values have arrived which means limit exists okay at that point by asking a question I'll show you, ok just take it easy and watch it You have to be patient after two-three questions You will understand yourself what is going on, it is ok What did Evaluate the Left Hand say first? limit of the function then you have left hand limit just leave you left hand limit I have to take it out okay so left hand limit now what did you say about the function The value is if x0 is equal to 4, okay? The value of the function is zero if x = 4 We have to find the coefficient of t at x = 4 what does it mean when the value of x is on the right hand side If I put it from side 4 then the function key What is the limiting value when I increase the value of x Value from left hand side to fur if yes then what is the limiting value of the function It will come but here I just asked you this left hand limit mat left hand limit means left hand limit means when you left going from the hand side to the fore when you Going to the four from the left hand side When you go from the left hand side to the four then what is the value of the function he is on four why are you not checking on four This is a big reason not to check understand the matter, look here brother if you are here If you keep four then a row will be formed below and you Do you know it is a sin to put a zero below in maths I'm just giving you a little gist right now what do i do i find out that this what values is the function returning around 4 ok but i need left hand limit in this I have to tell you, that means from the left side towards the fur if I go then what value will the function give now listen carefully fx4 negative 4 negative mode x-4 up x - 4 pay attention it is clear I want to tell you this thing, the value of the function When the value of x is moving towards 4 left what will happen from the hand side, what did i say If you replace x with negative four, then some four Put a value slightly smaller than 4 - h OK Now the value of x is 4, then the value of h Zero simple then limit h tends to 0 limit h tends to 0 instead of x just replaces x with 4-h So 4 - h -4 4 - h - 4 your mode is applied above come Let's find its value from friends four If four is cut then this is yours Limit h tends to 0 mode of my a mode of - a upon 4 to 4 This is yours going down - it's coming keep listening Now friends, yours is positive, understand that yours I told you that it is very small The number is very small but positive Positive Numbers on Numbers okay so tell me if you wrote this if this had happened then what would have been the answer to this if it was written like this then it would have been the answer If h is positive then it means minus inside your negative number mod your negative It removes it, you know the matter, mode off -22 mode of -33 friends mode of my a your h will happen then h up - a limit h tends to 0 So friends, your answer is -1 why from h I understood it was cancelled, now someone will say that if h's If the value is 0 then how did 0 get cancelled out? You guys know that in maths so if I h's If I keep the value below then it will get cancelled It is possible that you were taught this Bhai it is simple, if you cry then you will tell him If you cannot solve it, you cannot proceed further Then there is a row below, why are you cutting it? If the value of h is in the row below then you can cut why are you doing this friends, row the value of h no h if it was written equal to 0 then you would have done like this he could not do it, he said that Values are passed from row to row There is no exact row so I will cut here I am doing this, it is a very important thing There are small details in maths, so -1 comes okay friends so left hand limit did the -1 come means when I enter x in this function Key value from left hand side to four if we increase the value of the function towards -1 will increase I understood, I thought, what should I say next? what are you saying next Illustrations don't evaluate the right hand The limit is the same function as the right hand limit If you want to tell then basically you have to tell that limit and where to tell at x = 4 then limit x tends to 4 positive f x means x - You have to tell the value of mud of 4 / x - 4 what do you mean when it's from the right hand side I'll head to four So the value of this function tends towards what value I'm trying to tell you that it will increase How will you tell, look carefully at the limit what did i say 4 plus x to x The value of x is slightly greater than the value four Slightly bigger than four so made it 4 p h Now let's keep a limit friends h tends to it will become zero why does it become zero Always look at the value of x, something like four x value something like 4 then value something like h If you can imagine it like this then it is yours x 4 + h instead of x - 4 4 + h instead of x - 4 up is your mode 4 over 4 over 4 over 4 So your friends, the limit has arrived tends to 0 mud o h/h now h your positive hey brother mud off 2 is 2 mud off 3 is 3 If the mode of 5 is 5 then the mode of a positive number If the mode comes then mode is off h h h / h why cancel it brother if it was h 0 then it would have been I couldn't cancel it, you don't have h 0 h tending to row h h approach is zero h Approach me and I am tending to this. This is your one it is oh ho ho I understood it means you understood Can you find what is happening when you run the function It is the same when you hit the four key from the left hand side If you move towards then the value of the function will come to -1 when you hit a four from the right hand side when you move from the right hand side x that if you are increasing the value towards fur then this If the value of the function is approaching one then Different things are being revealed from both sides There are different things to be known from both sides if it is there then what will you say limit exist you would not do it if asked at that point what do you write limit x tends to 4 whatever your function is does not exist why left hand limit right Friends, your hand limit is coming differently But I told you this still, did you understand? Right now very basic things are going on so Learn to do questions 4+ h 4-h right hand Limit left hand limit removal kya bolne hai We are evaluating the left hand and the right Hand Limits of the Function Defined by This function is given to you Friends, this has been given to you and its meaning what is to be told at x = 1 tell at x = 1 okay what does x = 1 mean when you turn left The value of x from the hand side is one side when you press the x key from the right hand side If we increase the value towards one then the value of the function what will be the value if you understand the meaning friend once Comment what are you feeling friend You want to say okay you know too much I am putting this This is the same thing that will go on throughout the lecture. okay what are you saying friends The function given to you is LHL at x = 1 rhl tell also show that limit does not exists also shows that at x = 1 or else the limit exists at x tending to 1 no let's see okay now give me everything limit first of all i made the pen a little thin do it slowly i wrote limit x tends On two, neither negative nor negative means left moving from the hand side towards one, good which function is smaller than friends if the value of ex is smaller than n then this is used do i need to do this or use a greater medicine but this is a function between 0 and 0, this If the function is smaller than one, then it means that it is It is slightly smaller than the forest, which means we are talking about it If you want to use this function then press 1 x s I have written it here now what do I do I told you 1 minus so the value of x is something like 1 - h keep it ok so what is the limit Your limit will go and it will tend to 0 because if the value of x is one then one will go there h your zero will come limit h tends to 0 Instead of 1 p x you wrote 1 - square of h please solve this then it will reach the limit zero 1 psv squared and h squared a squa minus ka 2 okay well now what You can give it carefully now friends Put zero in place of direct A, no problem no it is ok instead of a direct g someone If it is no problem then here is your Friends, limit it to zero. Your 2 ps0 square -2 no. your a If it goes then your left hand limit will come to you If we look at the right hand limit then limit h tends to sorry limit x tends to from the right hand side to one so x1 Big means you will have to use 2 - x ok So limit h tends to direct row write I give you 2 - 1+ h in place of x 1+ h in place of x If you understand then how much is coming boss this is yours the limit h tends to 0 2 -1 - a if If you replace h with zero directly then Understand what it means to say that your forest is coming what happened brother left hand limit is coming The right hand limit is one and both limits are coming The beans are not coming so what will you say there For you since your left hand limit is not equals right hand limit there four limit x tending to f off One does not exist from both the places things are getting known so what am I I will say that limit doesn't make existence easy If you move ahead now, you will ask this same question for a second I will be able to deal with it but initially you will have to deal with this You will have to do the same question in the same way that limit exists shows that limit exists at point one so brother now it doesn't look like this here What was given in this question I had given that check at x = 1, this Check the given value in the question at x = 4 I have not given anything like this here but I have given it to you happened if you notice limit x tens to tending to 1 diya means checking on punt 1 means when the value of x is one to the left On the side of the forest, do not understand So friends limit limit x tends to 1 If I talk about negative then this is smaller than one The function is going to be 5x - 4 then here I have written limit x tends to 1 from positive one big light big light big so you this one we will use the function 4x - find out 3x both limits so you know that the limit what will you do, write here directly let h tends to 0 and 5 instead of one 1 - h -4 remember what I said, 1 in place of x If you are writing my h then the value of x is one so h your 0 so limit h tends to 0 x instead of 1 Solve this problem, your limit has reached tense two 0 5 - 5h - 4 if you replace h with 0 If you can keep it then 5-4 friends will come to your house If it goes then your left hand limit will be one look here same thing limit h tends to 0 x Cube of 1+ h instead of 3x - 1+ h instead of 3x ok, try a value slightly greater than one and solve it limit h tends to 0 this is your cube of 4 what is the cube of h h cube formula a hole cube of pb if you don't know Note here that the hole cube of a Pb is a cube p b cube p 3a s b p3ab is the hole cube of your a pb the formula is ok so the cube of no I did the cube of h and then what is 3 multiply by 3 multiplied by its square into this tov's square 1 h will come p 3 If m ba 1 then 3 * h s then this is your 3h s will go to minus 3 minus 3h okay now If you open the bracket etc. then it will be yours will limit h tends to 0 4 p 4h k + 12h + 12h You can directly put 'ro' in place of 's-3-3h h' give me what will be its value if you If you put zero then 4 -3 this will also be your one If it goes then the right hand limit is also coming to one If your left hand is creating limits then what are you doing you will say that since since left hand limit and Right hand limit equals one fight value There is a four limit coming, that particular one The function exists at the point I understood, this is what I said to prove it that the limit f exists at x tends to 1 It is clear and easy, in my opinion it is very much Easy, let's move ahead boss, do the same thing What is a question you are saying, watch it carefully now okay the function is given this function to you I have given that the value of the function is zero But two will come, there will be some function and in this manner performing an expression that if the value of x If I put zero then the function will give you two values and if the value of x is anything other than zero then the function gives you this value ok i told you show that limit doesn't At what point does age x tend to zero You have to check on zero, you will understand the meaning when you will go from the right hand side to zero when You will go to zero from the left hand side Understand that this is my left and this is my right in that way I am telling you that I am there for the function Let us see where the value will go Limit the first time x tends to 0 negative I am finding the left hand limit from zero Smaller than zero means equal to zero. If not, you will have to use this function x - mud x up x let's do it you know x The value of is slightly smaller than 0 micron if you write limit x tends to 0 negative sorry limit x h tends to 0 the value of x is 0 then h is yours too Instead of zero x I wrote 0 my h minus 0 mine a divided by your 0 mine a okay hey do solve this, this is your friends Went limit h tend 0 mine a i mine ay yours Minus friends mine a so this is your h positive is h positive then minus of positive means this Your h will come out and become mine a below So this is yours -2 up from my minus minus h to h this is your t so left hand limit your two has come okay limit x tends to 0 positive 0 equals positive 0 If not, then you have to use this function x - mud x / x let's do it quickly you all know that The value of x is slightly greater than 0 + h If you write then this is your limit x sorry The limit h tends to 0 instead of x is 0 + h - x in place of 0 p h upon here your 0 + h solve it then limit h tends to 0 this is your friends It's gone h h Yours has arrived - h why 0 + h h The mode of the positive number will come out There is a minus mark below your h is above Yours has come, it has come down, it is yours, this is yours I cried, this is your zero, so what will happen to you I understand that the boss has left Hand Limit and Right Hand The limit is not the same, is there a four on zero? You will say that the limit is at that point i.e. zero But the function key does not work asist it is the same thing that both cameras They are telling different things so what should I do I will say that limit doesn't exist easily At any point you can find the limiting of the function You learned to extract value, what did I say At any point you can find the limiting of the function Learnt to find the value of limiting value I mean, at that particular point I The value of the function is what you are interested in I am not from the left hand side when The value of x is going towards that right On the hand side when the value of x is if it is going towards the point then the function what is the value coming, let's proceed further over it and then after that I'll give you I will tell you about indeterminate forms And then I'll quickly tell you the limits and how to solve it easily right now Till then brother we are doing this with 0 - h 0 + h There is and easier way too but first Look at what he is saying in the question Suppose If fx is less than 1 then the value of the function will be a + If bx is equal to 1 then the value of the function will be 4 If the value of x is greater than one, then the value of the function The value b - ax1 fx1 what are possible values of a and b What are possible values of a and b aram first read the question, what are they saying if you understood this part No, I mean I was successful in teaching, brother. What are you saying till now, limit of someone The function is called that point but it exists why does it exist because that is equal to f1 brother f1 Tell me what is the value of f1 function What is the value of x at value one at f1 Brother, the value of the function at x = 1 is 4, which means You see that the f off gave you fur Do you understand the meaning of what has happened here what did you give me that boss limit I will make the pen a little smaller and give it to you is the limit as x tends to 1 the value of f x is equal to four what does it mean This means that boss, this means that Boss limit x tends to 1 negative fxxx.pro Meaning the limit exists here I have already told you that limits exist How did we know that its value is equal to 4 limit f x x tends to 1 4 coming means Limits exist, do you understand what I am I am telling you that he has already told me limit exists means left hand limit right hand limit is equal to fur okay Let's solve it, we might get some answer brother. Maybe let's do it together yes friends this is limit ok x trends to 1 for values smaller than negative one For values smaller than a you can use a + bx greater than one for values greater than 1 For the value you can use b - a for a multiple of 4 it is equal so we can calculate it It is very easy, smaller than one, you all know yes friends what will you do here you are limited Tends to you limit here h tends to 0 and in place of a + b you will write a word lighter than one you will write small 1 - h whose is it equal to Limit trends to 0 b - a is slightly larger than 1 + h4 is equal to let's solve it so now you Either put zero in place of h directly or It's the same thing if you come here directly If you write the place, friends, this is your A + B Aa try this and see its value in place of h If it is zero then this will be your b - a okay this is yours 4 friends this is yours The equation is coming now you can calculate a and b from here how can you take out much ease from this brother What does it mean that these two are equal equal and these two are equal so what do you do You make the value of a p b four and you and you mine do the abp trick add both the equations So this is your 2b equals 8 so b You got four if you put four in place of B If you write a p ch equals 4 then a is yours Friends, zero has come, this is your answer B.C. A0 B Ch A0 do you understand this point should the limit f a extends to 1 4 is equal to means the limit exists Meaning left hand limit right hand limit Both of them are equal to four, very easy Very very easy come on boss let's go ahead ok now finally let me tell you that I am speaking for so long, indeterminate form indeterminate form what is this There is a boss, what happens is that the boss sees you You will understand that camera thing, brother. If the cameras are turned off at the point then I will look around I am wondering what is going to happen with this Find the point by looking at the nearby points I am going to tell you what is happening at this point now what does it mean here sometimes Is the value function such that the limit x tends to Let's say you took 3, top x s - 9 and bottom it is written x - 3 now this is a function listen carefully ok hey friend understand if you put 3 in place of x then come down to zero will go up even higher if you look at the square of 3 90 has come up but even if zero doesn't make any difference It doesn't matter but look it has come down to zero Meaning you cannot get the value of the function at zero I can tell you that you know this very well Below zero is sin in maths so if You see something like this here brother what is happening look here 0 ba 0 is coming if you are keeping the value of x as 3 If yes then the value of the function is 0 or 0 Now what does 0 ba 0 mean in this zero by zero Friends I am an indeterminate form I tell you what it means, I am telling you like this You have a total of seven indeterminate forms in your syllabus currently in your syllabus This is what you need in your syllabus right now I am the only one who will show the form talk about it Now I will go into this in great detail I can tell you in half an hour indeterminately A little basic on what does form mean I'll tell you so you can catch it, okay, look Suppose I wrote 0 times 0 equals hum I believe that the value of Bhai 0 Ba 0 is not for anyone it will come okay so should I set this as 0 equals 0 can i write this to times 0 I can write it equal to 0 times yes now you Tell me everything, just think about it in your mind that what is the value anything can happen to my brother If you write 5 in place of , it becomes 5 * 0 in place of . If you write 6, it becomes 6 * 00. Can there be multiple values? Multiple Answers either you do it this way or else You do it in such a way that the boss's boss becomes 0 ba 0 which is k multiple answers are possible yes or no Multiple Answers K Multiple Answers K It is possible, okay and friends, understanding means 0 Multiple answers for Ba 0 are possible that's why This is called indeterminate form, similarly There will be multiple answers for this as well remember this also this also this ok so by indeterminate form you mean I understood boss multiple answers come Multiple answers come, so that is why I speak the indeterminate form and That is why instead of checking at that particular point I check around it's clear okay now what am I saying here one more very important thing okay He will say that brother the power of infinity now if If I ask you, what is 1 to the power of 100? The power of one is one If I say to you 1 If one lakh then how much is one Even if you power it, it will come out to be one, then Brother, you are considering the power of infinity as one If only one power infinity is coming then what does it mean There is only one possible answer, multiple There are few answers, I will say it again what is meant by indeterminate form If multiple answers are coming then type one Why is the power infinity indeterminate form It was said that this too has answer one Will you understand what I am saying to him Indeterminate form is said because here But wherever you see zero and one It is giving you zero here wherever you are And the one that is visible is this exact zero and there is no forest Exactly is not zero or one or is not one ag or it is not one it is tending to zero or one what did I say this is tending to tending Two means it is going towards zero or one going towards tending to zero tending to zero or it is a forest what does it mean to say listen This means that if the boss has the power of Exact One If it were infinity then its answer would be one if If the power of the exact one was infinity its answer is one but but but but but but but but but but brother This is not the exact power of infinity This is tending to 1 to the power of infinity tending to the power of infinity and hence You call its answer indeterminate because tending to the power of 1 is infinity Indeterminate The form is clear, it is for sure that all these All of these are your indeterminate forms Multiple answers are possible so I am going to Indeterminate form I tell you the syllabus I will get this now brother, now this method is mine have you seen a lot brother that brother limit To find out I will use the left hand limit I am taking it out, I am taking out the right hand limit h 0 - h 0 p h By doing all this you directly How can we find the value of limit also Listen carefully, you can take it out directly also how are there now two types of limits one is algebraic and one is non algebra is ok if there's a problem on Limits does not evolve trigonometry Ewers Trigonometry expo cial expo cial logarithmic function then it is a problem on algebraic limits otherwise it is a Problems on Non-Algebraic Limits Simple c thing if the function has trigonometry or is it an exponential function of e is a logarithmic function of power x ok or your evers trigonometry it is a function if in a function you get all this things are visible means they are non algebraic limits would be okay and if You will not see these things in the function It means trigonometry is not visible sinx-cosx You can do factorization if possible If it is not possible by other method then do rationing You can do it or by using some standard limits or else write this and cross it out Friends, this is none of your business cut it off you guys okay so basically Broadly speaking you have four methods of meditation listen what I'm trying to say okay First of all look carefully at the question now I asked evaluate limit what does this mean that x = 1 doesn't matter to me at x = 1 But I don't care if this function What value will I give to this? I am interested if I can participate in this function should I go to one from the left hand side or If I go from the right hand side towards one what value is the function returning ok so what can you do brother left hand limit To remove 1 - h, put it on the right side in place of x To remove hand limit you will have to place 1 p h And then you will see brother's left hand limit Is the right hand limit coming yes or no Now you don't have to do all that, what should you do if If you are able to do a direct substitution You can do it if you look carefully x e 1 can you put it here directly if If the indeterminate form is not being formed then it is very It is a good thing, if it is happening then you There are a few more things to be done, let's just see go to limit x tends to 1 you replace x Write directly one, how many answers are coming 5 P 4 9 P 3 your 12 9 P 3 your 12 has come do you know what it means, this is what it means that when you do that thing, brother you are left handed Even if you go from side to side then 12 would be right Even if you go from the hand side to one side, you would still get 12 Or think of it this way, when you do x instead of x you write 1 - h when you write 1 + h even if we write it and then calculate the limit value I directed that she would come at 12 only. Direct substitution is okay If direct substitution is done If the indeterminate form is not being prepared then do it ok, i told you to remove the limit, this is enough If you do it well then limit x tends to 2 I can put 2 brother 2 + 3 down 5 and 2 - Sorry 2 - 4 friends of sorry 2 zero So what is your 2 = 4 / 2 + 3? friends this is coming your way Yours is coming friends 0 ba 5 in Is the determinate form being made or not? The remaining amount can be zero so this is yours Zero has come, which means you are understanding this Left hand limit right hand limit of function Limit at pot is function key left hand limit right hand limit at point to zero Come on, do you understand what I am saying? I understood the concept again and again article last time what should I say when you come to this function in this function the left hand side towards two will we move from or towards to the right hand side If we increase by then both the sides will have same value That will come, it will be zero, you should try it as well you can use a calculator okay let's see now Like it came in front of you to see it comfortably it is easy first direct Substitution is the only question so ask whether x = 0 If we keep it here then the forest is coming down a lot great no matter what it is Look up brother, I am crying above, so I am under the root 1 1 a √ 1 1 so 1/1 friends this is your two a there is something you can keep then directly If you can make substitutions then limit it. x tends to 0 a √1 + 0 + √1 - 0 is your 1 + 0 So friends what is your answer? a √ 1 1 a √ 1 1 down here your forest so this is your 2 has come, direct limit answer 2 has come Can I do direct substitution? Look carefully, the value of x is 2 if you If you put it here, 2's value will be 4 4 - 4 0. Your zero is coming down, 2 s 4 4 - 4 - 10 - 6 friends, zero is coming on top as well If yes then it means 0 ba 0 form is coming Indeterminate form is coming what kind of form is coming in determinate form if indeterminate form is coming then you Direct substitution is not possible You will need to remove the indeterminate form What did I tell you about this form? I will have to remove it, what will I have to do to remove it Do you know how to remove it and what to do? there are ways to remove there are ways to remove These are the methods of factorization Easement by using some standard limits I'll tell you what it means, just watch it look at me comfortably go okay pay attention okay now these Determinate form i.e. 0 Ba 0 form will be removed How the second method is factorization If you look carefully, I have written it as limit x tends to 2 tell me up i am in the middle Using term spitting I can write it like this x s - 2x - 3x + 6 up to x - 4 you can multiply the x by x - you can say 2x + 2 this is what it is called factorization so above your friends x common Then, x- Above your -3 common then x-2 or your basic The maths is below, your x-2 and x+2 here your limit is one ten to 2 let's see so the limit x tends to 2 from above to x-2 out x-3 left down your x-2 a p 2 left now listen carefully x2 is subtracted from x-2 now Will someone tell me how did this get cut by Boss x? If the value is 2 then 2 -2 0 2-2 0 ba 0 Boss friends how did this x tending happen tutu is tending tutu x's value is not to okay so this is cut now because of which 0 By 0 form was being made, it has been removed now you Friends, directly by keeping value to it Look, the indeterminate form would not be getting formed. Look, if you put two below, four comes up as two So if -1 comes then you put it directly x my 3 Up x+ 2 put 2 directly in place of x so 2 - 3 up 2+ 2 so this will give you - 1/4 So what I mean to say is that your limit has come to - 1/4 In this function when you are on the left hand side If you move from one to two then your function is - will give the value towards 1/4 and when you click on right Even if you move from the hand side to the two function will give you -14 value easy this They say factorization, now look here. what is happening on keeping one look again 0 ba 0 form is being created if 0 ba 0 form is created means you are doing factorization rationalization Going into all this can now be done here Factorization See How the Limit x Tends to You can write 1 x c - 1 as x - 1 x s + 1 - x You can below your x-1 hey brother I forgot Look at the identity, here a k - b k is a mother asp a pb sk so do that i see aqua squa psv my a sorry ya plus p everything will happen plus it will be clear now Was this the reason why these terminated forms were coming If it is removed then your limit will be x trends to come gone up your x sp one now you can see the x Place and store 1 square of your 1 PW Pw friends if yours came then limiting The value of this function came to 1 3 and the same goes further Look at this, if you put 1/2 in place of x, you get 0 The form is being made, try it Direct substitution can be done to form 0 or 0 it is happening means indeterminate form is being formed You will have to remove it, see how to remove it. Here limit x tends to 1/2 8 8 x - 1 friends I will divide this by 2x Hole Cube - I can say the cube of 1 and Below I'm going to multiply 4 to the power of 16x by 4x. Square - I can say square of 1 correct then limit x tends to 1/2 up your if a - b is added then a - b is added equal to a - b a square of means your square of 2x is 4x the square of 1 is 1 plus plus both of these If you multiply then your hole will come 2x Divide by a s - b sk is a s ma b sk I a- b a p b can be further factorised below how to limit x tense to 1/2 2x - 1 4x Scan 1+ 2x Again you can convert 2x into 1 square square so this is yours this is yours 2x - 1 * 2x + 1 Instead of this I wrote a+b*a- b Now see the reason why indeterminate form comes 0 by 0 form was coming, it got removed Now you can store your value directly So limit x tends to 1/2 you direct put here square root of 4*1/2 = 1*2 , 1/2 divided by below your 2 * 1/2 p 1 And here is your square of 4 * 1/2 You should understand a little bit of writing, otherwise it will look It must be happening I think yes it is visible So friends, the square of 1/2 is 1/4 so 4 Friends, after cutting 4 from this, your one will come Isn't it true that here you have been given a forest by friends and it is divided by 1/2 2 friends this is also yours If only one comes then your 1 p 1 p 1 comes above went down to u went 1 pv your 2 has come Here your friends 1 pv this is also yours If you go then basically the answer is there 3/4 ok easy let's go ahead again ask the same question here also see if yours If you pay attention carefully then what kind of form is being formed? this is also there friends if you look carefully Zero is being formed below, if you see the hole then you will cry Ba 0 form will come again what should we do Be careful that you don't keep the direct one if you can then limit x tens to what have you done Karv for x s you wrote 1 - x in 1 p x this is your 1 up x - 1 any The problem is now pay attention to what I am doing here I am here okay what did I do I wrote this two here I have 1 my one okay here I have 1 p x is here friends I have used this plus I put a minus on it, you think so, I did it above multiply minus by minus below minus from above to minus so this is your -1 from below to minus So friends this is your minus below so this is p This will become yours - x will now become like this Why did I do it so that Dino could do it in 1 minute - x 1 - x you see the same thing now you know how do you solve this thing lcm Just basic maths from LCM to anything else There is no limit x tends to 1 for both of them What will be your LCM 1 - x * 1+ x what will come above is written same so above here pay two will come minus here put 1 - x is 1 if x is not one then one By going into this your 1 p x will be multiplied Solve it and you got limit x tens to 1 2-1 - x down here your 1-x not 1+ x now just If you notice, 2 - 1 1 then your 1-x will appear above. friends 1-x will be subtracted from 1x which Indeterminate form was coming due to reason When that part of yours got removed, yours came here limit limit x tends tove limit x tends two 1 up p a ch will give you 1 up 2 now You can directly replace one with another ho okay let's go next what are you saying question I'm a great question if you pay attention again. Friends, this is your zero by zero form. It will be visible, check it, instead of one So 1 minute below zero okay look at this also Zero will come down, in this hole you will see 0 ba 0 only it will come okay well it is coming down to zero leave it What will you do in these determinate forms? You count it, this is the way to solve it I'll just make the pen a little smaller limit extent comfortably see ok above your x-2 is down to x s my x I got x If x-1 is left then x is not x then factorization I am basically doing this if I x If I write common then here x s e 3 a p 2 Middle term sting can be done here If yes then do it, then this is your limit x tend tov x mine 2 xin x Mive 1 direct access to your phone This will come to you friends from my side x my2inxmy2 should have checked it once do you have a smine 2x mivo xp 3 plus flat sorry your x will come this way is correct ok from the middle term spung now look at x in x-1 is one so when you take the LCM you will easily will come x will come x my will come x-2 will come ok Now if you have written all the three things here then on top your myway as it will come here what then x in x-1 goes to x-2 then x-2 in x-2 ok this is your limit x tends to 1 Solve it, here is your limit x trends tove up if you look carefully up if you If you look carefully, this is your one second something Was there any mistake x * x - 1 okay this is yours x1 so this is yours one second one second one so this is your one upon Zero will come here, something is wrong it seems to me that something is wrong why is it appearing like this to me Because it's not cutting or anything right now no no its okay its okay I multiplied what I was thinking x s - 4 is being formed but this is x-2 x-2 so Here's what you can do Friends, here comes the whole square of x-2 -1 Below your x * x-1 * x-2 light calculation and nothing else is ok so this Your limit has arrived x tends to 1 limit x tends to 1 x - 2 to x + 4 - 4x-1 up to your x * x - 1 * x-2 If we simplify the above a bit, then x limit x trends to 1 up your result will be x s - 4x + 3x*x-1* x-2 just above again middle term splitting Same factorization so limit x tends to 1 limit x tends to 1 ok i don't understand If it comes, you can easily do it right Think of two numbers that can be multiplied If I do that, I get th. If I do that, I get -4. I think you will have two such numbers why -3 and -1 because Minus no point No, brother I am bored, stuck sec x s - 4x + 3 you can divide this by x s -3x - 1x + 3 If it is correct then we can write x * x-3 -1 * x-3 so x-3 * x-1 is your result x-3 * x is your result what do you write above what do you write above x - 3 * x-1 down is your profit x * x - 1 * x - 2 Is there a reason for indetermination The form was coming, it has been removed, now you can directly If you can keep the values then limit x tends to 1 x - 3 or you can directly calculate 1 -3 1 * 1-2 what is coming value 1 my 3 is -2 this is your -1 so this is your two Your answer is here Answer 1 - 3 - 2 is here Your to has arrived, okay, let's move on now See how these ones will be, watch them carefully okay now what I learned is that brother if Indeterminate form is coming so I I can do factorization but here under Route is engaged whenever under route is engaged First of all think that direct Is substitution possible? x tends to 0 So here I will write 'ro', if I put 'ro' above it then it will be √2 - √2 so 0 ba 0 form is coming to you 0 Ba 0 form is coming whenever 0 Ba 0 Form came, what should you do, go to methods What method is factorization taught here? Nation celebration will not be possible here If possible do it then limit x trends how to do 0 a √2 + x - √2 up x multiply up and down but who gets the same thing just put a plus in the middle then √2 p x p √2 √2 p x p √2 Solve the above Multiply among themselves the lower one among themselves You already know the multiplication above, friends. √2 p x √2 p x √2 √2 minus plus then a- b a p b will be equal to a sm mine b s that's what's gonna go down Below is your x in √2 expression: a p √2 and this is your limit x tens to 0 what came up is the root of 2 p 1 If it is square then it will be cut on your top just 2 ps x will be left here your only two your rescue is here under root 2 in u bracket one p √2 this is your limit x tends to zero is okay two is cut from two okay someone no problem no problem no problem no problem boss no problem no problem no but one second √2 ps yes no problem no no problem no problem No problem, okay, now just look carefully So here you go, limit x tends to 0 up comes x down your 2 multiplied by √2 + x + √2 which is why the terminated form remained It was gone if you look carefully So above if you replace x with direct zero If you keep it then there is no problem with the zero above is below if you put zero then it will be below Zero is not coming very good so now you If you can do direct substitution then this Friends, your limit has arrived x tends to 0 Your 0 on top is your 2 below √2 + 0 is √2. your + √2 so this is your 0 * √2 + √2 2 √2 so 2 * √2 anyway yours came yours came 0 bye 4 √2 anyway your cry comes up your Zero ok friends this is your homework this will be your homework okay brother you have to release indeterminate form You must be seeing it, now let me tell you something I am giving you the standard formula brother Learned to do direct substitution Did you learn factorization or even ration ei jhana? I learned that these three things cannot happen If so, then is there any standard You can also overcome limits with the help of Formless You can evaluate the answer of limits Can you tell me what is the number one formula? Paying attention is the number one formula that will help you Do you want to remember that boss if the limit is x Tends to have a listening limit x tends to a is ok x^n - a^n / x - If a is written then its direct answer would be is n-1 you have to remember this directly How come the first formula is not in all of them I want to go directly, remember the formula, now how You will recognize him, just take it easy, it's ok The first thing you need to know is the power of something a to the power of something x and a to the power of x something a to the power of something x and a then down up also x mine a down also x mine a just Above is the power of x and a this is very important limit x tends to where x is going Value friends, this number is written here It must be this place where it should go The number is written here, you should go now I will give you an example, don't worry, what is the answer This is the simple answer, what is this power that is involved Power multiplied by this into power in one Reduce the power multiplication by numbers into reduce the power by one now give one on this Let me ask you four or five questions And after that I will take you again Other formulas are based on trigonometry what did he tell you, he told you limit evaluate now if you pay attention first Direct substitution is the only option Direct substitution is not possible If you keep it then you will form 0ba 0 Direct substitution is becoming a 0ba 0 form Try it, there is zero on top and zero below now if you pay attention if you pay attention What is factorization without applying rationalization Something like this is visible to the power of x Something minus number x minus some number and x Tends to 2 so should I take it to the limit x tends to 2 can i raise x to the power 10 let me write 1024 to the power of 2 as 10 to the power of 2 why is it written limit x tends to 2 I just told you I just told you was whatever limit you have where x where But he is also approaching here, the same thing is going to happen I want a a a a a if you understand then 2 2 2 2 2 2 and Below is your x-2 and I told you the answer What power? Write the number. Power of the number one. reduce it, okay then this is your 10 * 2 10 * 2 raised to power 9 you can solve 2 to power 9 512 This is your friends 5120, understood 5120 has come easy then direct standard You can find out the limits using the formula Look, it looks the same, look at it You write this as limit x trend power 10 of x is your power 10 x to the power of 5 minus this is to the power of 5 You can write it but it looks like there is a problem not giving in the formula in the formula There are powers above and below but no powers below is above is to the power of x something then below x to the top power sum down is just what you can create it, how do you create it pay attention what did you do what did you do Brother, are you the one above it or are I below it? x - 2 and below that also he asks for x - 2 asks for yes or no, yes or no, what do you understand what am I saying do you understand what am I saying again I will tell you brother if the power of x is 10 - the power of 2 If x - 2 is below 10 then it is very important for me. well that's fine if x is to the power of 5 - 2 x to the power of 5 would be - 2 so for me that's great what did i do above Divide by x- in both the places below I can do this to you right now I will teach this thing in Algebra of Limits But you understand brother numeration denominator You can give different limits in Basically your limit will become x tends to 2 x to the power of 10 -2 x to the power of 10 / x - 2 Divide by limit x tends to 2 this is also I will tell you, you can give different limits are x to the power of 5 in the numerator denominator -2 to the power of x2 is correct now its answer You will know directly what power will come reduce one in power what is the answer reduce the power in power u friends yours has arrived Divide 2 into the power of 2 into the minus 4 so 2 to the power 5 32 * 2 will be 64 64 aa will go, I have understood your answer to the limit, okay let's see ahead let's see ahead well first you should take a look at this once Just see it once so there will be no confusion further yes okay friends this is going to give you some algebra of Playing within limits is basically playing within limits How to properly note this topic Algebra of Limits Algebra of Limits Hey friend, what happened, understand it yourself friend now the body is stuck brother wait a second Come on, what are you saying, pay attention brother, limit x tends to function plus function if it is written inside then you in both the functions You can set different limits, okay brother is the whole limit of the function minus the function you You can put different limits in both of them and you can minus it ok brother this is very it is important is it very important As if Constant brother Constant is also a Function is a constant function is a constant multiplied by function then you take out the constant limit only you Put it in the function, did you understand what I said? I am doing it okay see what happens next brother function not to function hole limit is applied on you You can give different limits at both places There is no problem, it is just a simple formula no division also same thing brother if Numeration means function upon function so You can separate the numerator and denominator of the function I can give different functions which are already I had done it, just remember the one below You should not be zero, you should not be afraid of it That means your zero should not be below okay and what does that mean is very simple It won't work for you much brother If a function is greater than or equal to another function is equal to the denominator of the value of x for every x This function for a value greater than or If it is equal to this, then its limit is also equal to that The elder one will come equal to her okay let's go to the next question what are you saying this one now see if its the same question can you see it, look what is this limit do you need limit x tends to a Do you know the limit x tends to a power of x a to the power of a up to x - a in the form It is visible how is n-1 visible how should it be visible I kept telling you, look here there is something x here there is something a here p something x here p something now you look up if you look up you will see limit x well by the way this is your itinerant form It will be made 0 ba 0 you can check it ok up guys your x*x ki pa 1/2 a * a to the power 1/2 below your x - a little So if you solve this, the limit will be x tends to a this is your x's pa 3/2 - a's The power 3/2 must be the same power as x - a It looks exactly like that, look at it x tends to a is here also a a a a a a a a down is aces minus down is x minus up You know what your answer will be power number power number and then one to the power If you reduce it, your 3/2 A power will become 3/2 my will be 1 but this is your basic 3 The answer to the limit will come soon, it is easy Look what you are saying, good question, good What is the question boss saying Look carefully, put your star in it Put a star on what you are saying in the question in question that look at limit x tends to 0 limit x tends to 0 x + 1 to the power of 5 - 1 / x now the first thing is the 0 by 0 form why brother are you coming down crying, come down crying It is crying down below, it means you will get changes We will have to do factorization, rationing or then if you carefully look at such and such things See if I replace x + 1 with y look at the thought if I replace x with 1 by y now this is your It won't come to your mind, but it's the first time It will start coming slowly, have faith, if you x Write y in place of + 1 and see what happens now. writing in y instead of x replace x with y If you are going then limit yourself to a trend instead of something y tends to be something what did i say if you use y instead of x If you are writing limit b then y tends instead of x If it becomes two then the limit will come here y What will happen trends, look at the value of x here If the value of y is 0 then do you understand this? let's see again the value of x is 0 x If the value of y is 0 then the value of y is 1 then the limit of y tends to 1 now instead of x + 1 you wrote y y to the power 5 -1 what will come in place of x The value of y - 1 I will say it again brother, use y instead of x + 1 above Below, in place of x, y -1 1 is the value of x, y -1 Now isn't this in the same form to you? you can see it here you see 1 Power 5 Felt take where the value of y is going to It is written here where the value of y it is going the same as written here y power something - 1 to the power something y - 1 What is the direct answer to the power of a number Power one less so this is your friends five answer The answer to the limit will come directly in five easy look next this is also a good question what do you say It is very easy and very easy too okay now look j tend to say it now Now now now it is the same thing, is it the power of the root can you apply something minus to the power of wa but down below you have to have a hammer you should have your jam here also So the answer will come in that way also but I what will I do just pay attention what will I do, I just did it power 1 or 6 is considered as one problem i have seen 1 thing in power of z Let's assume X, what is the benefit of this now? look the value of z is one listen the value of z one is taking x in place of z one is taking x in place of z I have been there so z tends to something now x If there is a tendency to something then instead of z one will be there Power of one: No matter what you do, one will always go away x of value and then this is yours limit x tens tow is coming now to z If you wrote x to the power and ba 6 then below is your x -1 to the power of z 1/3 what happens if you Squaring both sides here If you square both sides, then z The Power of 1/6 of the hole square equals x [ __ ] okay z to the power 1/6 of guys what will be the square of z to the power of 1/6 What will become of the square? The power of the power is mutual I multiply friends this is your power of z 1/3 will come to the power of x That means instead of 1/3 power of z write x s x S-1 like this now its answer is directly given to you do you know look 1 1 1 1 to the power of x something The power of 'va' is something direct answer to this friends the power of the power number is one less so this your two answers have come will you understand how to use limits I think that the questions need to be solved ok now i have told you only one thing till now Formula Explains Standard Limits of x Power of n - a to power of n up x - a limit ax tends to answer a directly to your n-1 You guys come within trigonometry limits I will tell you two or three formulas that you will remember right now. I will tell you two; I will prove the third one to you. I will do it because it will be useful to you in the future okay just pay attention to derivatives friends friends friends first of all The first formula is that the boss limit x tends to 0 sinx-cosx rho means whatever is the angle of sine is the x angle Whatever the angle of the sign, go towards the row Whatever the angle of the sa, it should be towards the row Whatever tan0 is going towards zero, it should go And the angle which is written here is required The angle should be written here It is written here, it should be written correctly Let me ask you a question for example question came limit limit x tends to 0 sin2x / x what will be the answer to this now look at zero by 0 There is a form brother, first of all direct You have to think of substitution only if you want to live So x0 is 2*0 equals 0. So this is your 0. Only 0 by 0 form is coming if 0 by 0 form Standard limits are coming, why impose them because your trigonometry is coming bro Now you know that the boss's boss is the first The point here is whatever angle it tends to be to be zero whatever angle you want with sign here it is written 2 3 4 one 200 whatever is written there your tends to 0 should be x tends to 0 So here x is written as the second important The thing is the angle here should be below If you want it, it is easy, first see it here Friends yours should be 2x the angle is 2x comfortably the angle is 2x so this 2x tends to row should I call him my brother if I will multiply both sides by two So this gives you 2x 2 * 0 0 so 2x tends to 0 what's the problem so whatever The angle is 2x tends to 0 and this is the second angle. what was the important thing that whoever was here there is an angle it should be down also so below it's just x you multiply by itself now If you multiply by yourself by two then divide by two also took it up down to multiply now you know what is the answer to this thing what is the answer to this thing limit 2x tends to 0 sin2x / 2x If the answer is one then 1 * 2 is yours friends Come on, you understood what I am saying Do you understand what I am saying, you look at it like this You can also see it in Algebra of Limits Did I teach you that constant If it comes out then 2 into limit 2x tends Two 0 sa 2x up 2x its answer friends your one If it happens then your answer has arrived I will ask more questions gradually and we will ask the question this has already happened this is done and then look what he is saying I am talking to you about sin3x / 5x, now it is fine Everyone knows that 0ba 0 form is coming Is it clear that 0 ba 0 form is coming yes Clear if you do direct substitution 0 ba 0 will come now what should I do pay attention I replaced friends limit x tends to 0 I made it 3x tends to 0 did I what is the problem what is the problem can i do it naa boss dekhna x tends to 0 when 3x Tends to 0 It's clear, okay, now here I Write down 3x and you have 5 of these x as a result should I do it down to myself down to myself Multiply it and then increase it further You will have to multiply by 3 below What is the advantage of multiplying 3 above 3 3/5 3/5 You can take 3/5th apart and it is yours Limit 3x Tens Two 0 sin3x / 3x angle itself tends to 0 down to 3x 3x its answer friends is one so Direct yours 1 * 3/5 3/5 good one arrived And the main thing to remember is that Boss Limit The value x tends to 2 is written here u It is written that any constant function is its Answer two will come, meaning the same constant The answer will come so I am using a constant function I don't understand why I remove the limit okay what am I saying see further what are you saying evaluate it is very easy All from NCERT It is a very simple question, what are you saying are just limit x tens to 0 okay sa One more below friends I have given different limits From the Algebra of Limits to Tendencies 0 sir now please be careful and clean it a little I will direct you, should I apply this? Brother, I can multiply from both places yes so a a * 00 both places from b I can multiply yes so b a b * 00 Why do you do this, whatever the angle is, it tends to Whatever the angle, it should be zero, it tends to two zero should be now a one with a down So you should multiply a by itself turned b down with a b a so here b You multiplied one by yourself, understand? What am I saying, it's clear boss ok now pay attention to one thing brother this x you can cut this x out, no problem it's nothing to worry about ok a ba b bacha If it is a constant then brother you will get the answer I know it and it happens, its answer is also one direct answer your a up b aa will go ok the correct way to write it what happens is that a gets b out first take out the limit write this limit Write it down, okay easy, let's move ahead and see if it works what should I do if I go brother look at 0 ba 0 form So it is happening because 0 and 1 are 1-1 0 So this is your 0 by 0 form coming what should I do now here I am I only use you as sa and na I have explained the formula, now pay attention do i know the formula of cos2x cos2x it 1 - 2 sa x is this formula you guys do you know now tell me if here How to understand the formula, just look here okay so this is just 2x so x here is 2x so here x if it was cos4x here then here But no matter how much it is, brother, it's his You can sleep half as much or you can also sleep like this Can you buy this product at double the price of this product? Double 2 Sas Your Basic Trigonometry This was the thing that the one who doesn't know this thing Go to Trigonometry and revise it again do it okay if you believe this but I just said Think about it and see what will be the value of writing okay look what did he say its double its double so whose double will be one or you will be like this You can also think of it as half of 2x x half of 4x 2x half of x is x times 2 so this is your 1 - 2 percent squa x ba 2 i remembered these basic things basically i am changing ka to sign okay let's go this basic one this one Just remember the formula, this one, this one You can make everything else yourself now here Pay attention to what I thought limit x tends tattoo 0 ok limit x Tends to 0 Excellent 1 mic x up x s 1 mic x what can i do I can write 1 My x What can I write or you can directly change the place of x then instead of limit x 1 minus x, we get limit x place you 1 minus 2 sign sc one ba 2 your just keep looking at the x s below okay brackets open then limit x tens 0 1 -1 p 2 cy s x ba 2 own x self served So here you go limit x tends to 0 2 If you want, you can take it out, it is yours s x ba 2 up x sine aa gya an angle of san what is the angle of the x weight x2 please pay attention Sir please pay attention I am out of limit outside the limit comfortably very comfortably okay Friends, instead of x tends to 0, I will call it x If I change ba 2 tense to 0 then there will be no problem here But I put 2 limits x ba 2 trends to 0 Divide both sides by two, then x / 2 0/2 0 it happens ok why do this sa ka The angle is x / 2 I have written this as x = 2 * Why did I write x / 2 because twice the sc x / 2 means x / 2 * x / 2 holes Divided by this is your x I will write s x s as x * x What am I doing, I am doing this am that I want that sinx-cosx and I will show you by writing it batu ten to 0 ok your sign is one Buy a multiple by yourself and keep the weight I have given him a separate limit, Can I give it a separate limit? Your limit is equal to 0 limit one by ten to 0 you move away Thank you so much, I can see yours here too x ba 2 up x now what have I done to myself If you put two below here then put two above also You have to put it here below or here above So basically I have 2 in the denominator multiply to divide multiply to divide What is the benefit of this? Pay attention to this thing what is the answer what is the answer to this thing Similarly the answer to this thing is one green So friends, this is your answer to the part one. I went and you can see two below here yours has arrived na, you can see two below and If you are seeing two outside also then its effect is Batu has come, Batu direct answer is clear Whatever the angle of the limit sign tends to zero sine of angle upon the same angle sign of angle is same angle okay right right larrat larrat next Look, it is a good question and it is very easy too. It is very easy also, let's see what are you saying 1 - cos4x now tell me one thing i have learned about cos2x what is the formula 1 - 2s s x just like this By looking at the formula you can change all the formulas. can 0 ba 0 form is being made fine Isn't it, check it out friends 0 ba 0 The form is being created only if you press x directly If you write 0 in place of 0 then K 0 1 1 -1 below 0 K 0 1 -1 upvote ok so this is yours limit x tends to 0 1 - cos4x 1 - cos5x Come on, I put 1 minus or you go ahead, okay? instead of 1 minus cos4x I wrote cos4x instead of 1 I wrote 1 - 2s 2x sa ka Angle brother instead of 2x cos5x it is simple here don't be afraid if i take cos5x here whatever Here half of it is 1-2 sasc whatever is here But his salary is 5x 2 Okay, right? Those formulas should come, right? So 1 - 2 cy square angle in 5x ba 2 okay now now the direct limit is a little bit I will direct you understand 1-1 cancel If we understand minus then the bracket will open then -1 plus -1 plus then 1 -1 can you be your friend 2 sa s 2x will come and below that your 2 s s will be 5x ba 2 will come cut 2 from u now this too I will just direct a little bit, okay? Like 2x sin2x sa 5x b a 2 sa 5x b a 2 brother here You multiply by 2 and put x tends to 0 You can multiply 5/2 by 5/2 x 5/2 * 0 0 so 5 x over 2 tens to 0 means whatever There is an angle here, you can make it here If yes then there is no need to write it now will direct now you know now you You know boss, there is 2x under sin2x. If you want then just think about it Pay attention, you should put your mind to the limit x Tends to 0 put 2x under sin2x put 2x over So you will have to multiply it by 2x also So put 2x under this 2x so put 2x Then you will have to multiply by 2x and divide it too If we put 5x b 2 below 5x b 2 on the left side then You will also have to multiply 5x by 2 Similarly, under sin5x b a 2 you have 5x b a 2 So now you will have to multiply 5x by 2. Do you all know the answer to this thing? forest of this also will come forest of this also forest of this also what is left of the forest 2x This is your save Limit x tends to 0 2x look x even one cut If even one x is cut then 2 * 2 becomes your 4 Your 5/2 goes down * 5/2 that is 25/4 up If you put one then limit friends directly Your answer is 16 ba 16 ba 25 aa I got it, you can do it easily in this way Trigonometry What needs to be done is to solve the limits, just solve it It is okay to do so, let's look at its limits now. But friends, I will ask you five or six more questions. And then we move on to derivatives Let's go, we still have some more questions left No, this is your homework, exactly the same question The last one I got you to do exactly If it's like that then I'm giving you your homework. yes this is the hint brother what is cos2x The value is 1 my 2 bits x So mx1 - 2s, whatever the angle is here. Whatever the angle is here, it's half of that. mx2 mx2 did you understand it, okay let's move ahead, it's very easy Just take a look at what they are saying now First of all think about direct substitution okay if you know the value of x see what comes so c pa know what how to remove it Guruji cos10 I forgot hey friend what is the use of Pa has to be a friend 180 okay degrees brother this is in radians So cos10 ° wrote cos10 you cos10 You can say 180 - 0 or 180 + 0 It's the same thing isn't it, which quadrant is 180 - If K is negative in th second quadrant minus put 180 pe k ka co to k 0 k 0 1 So -1 has come friends so your one above is this Min 1 mav your top zero is ok Let us see below what is the value of na pa that means do 180 just like a child I am here this is yours take 180 p 0 or 180 - 0 doesn't matter first of all Quadrant 180 P Theta Third Quadrant All Schools to No in the Third Quadrant If positive then positive will come but no change at 180 So tan0 of 0 tan0 friends comes to 0 okay So above as well as below, that is 0 ba 0 So this is your friends 0 ba 0 form aa It's going great 0 ba 0 form is coming so brother trigonometry limits standard If I have to follow then I will be in I know how to play I know how to play in sa hmm okay so what did I write to c x You know cos2x is 1 - 2s x so cosx–sinx 1 mi 2 sa s x ba angle of 2 sa x ba 2 ok just do it limit x tends to pa 1 p k x instead of k x you wrote 1 instead of . minus 2 cy sc x ba 2 and down t sc x in place of did you write sa s x up k s x what The problem is okay let's look forward but friend but there is a problem that brother this is yours X tends to get x tends to be found here your angle Tends to zero should be here now you Can you understand what is happening, boss? This will come to your mind that brother I told you yet bro cos ko sign na ko sign change the sign and we have the formula It is available but brother whatever the angle of the sign x ba it should be tends to zero right but there Whatever the angle is brother, please make this p x bat If you take that tendency to become a problem then Do you understand the angle will be ten to two times Zero angle tends to zero, so No matter what you do, you won't be able to figure it out now Now understand how to think what should i do then another thing comes to my mind I if I don't know the [ __ ] in the cy [ __ ] I should change my screw and see what the benefit will be. ok limit x tends to pa 1 p k akan squa x i wrote hone [ __ ] access into your action is correct now Ex attention limit x tend to pa here Your 1 ps a y your cos squa you wrote down x sine sc x 1 minus cos square one no problem okay now Look ahead, the limit tends to pa 1 p co x in cos sk x its 1 minus cos x co a s minus b score I am mine B in A P B friends, because of which reason Indeterminate form was coming and it was removed did you understand what i am saying limit x trends Two pies have you arrived? Now you can directly place your 1 mic are in place of a why do you know if If you put pa in place of x then the value of cos pa I have no problem with that I guess if cos pa -1 then square of mive upon cos pa myv tov minus of myv toy your 1 up 2 will come this will be 1 up Direct answer to limit now this is very This is a simple question, I will just tell you this I am explaining how to do it See, you know the algebra of limits According to that boss this is the limit x tends to 0 or you sign or you also have to 2x You can give plus this limit x tends to 0 You can also give this limit to 6x Tens to 0 you can also give it to 5x minus minus this limit x tends to 0 you say can you give 3x also how brother numeration I gave different limits in the denominator If you give different limit in numeration then There are two function additions in Newmate which are written in So its limit is different, its limit is also different If you can do it differently then you can do it now You know how to solve this, now you can solve it I know how to do it I will give you a [ __ ] from above Let me tell you how you will do boss brother here If you apply two then multiply both sides by two angle tends to zero how much has come here turn it down to 2x up down to 2x Multiply the same thing here, the same thing Now here what is its method number two Also see, I will not tell you method number two It's a bit lengthy, I'll tell you not to do it what can you do there you need the formula You should know sinc-sind You can also evaluate it using the formula anyway let's go ahead now look here What happened okay good question good question It is there in NCERT, I don't have any NCERT This is the question, maybe it is there, see what he says next Now see if the angle tends to cry Whatever the angle of your sign, it tends to If I don't cry, it means I'll have to do something else I will say it again, whatever the angle, it tends to be Two is not zero, which means you have to do something else So what did I think and I got this - x my problem is pam - x i have h I agree it's late, no problem, no problem no now you have to find the value of x listen to me Carefully find the value of x in a way that tends to pa then the value of x if pa then The value of h 0 brother look x if your father then it is your cry Do you understand what I am saying x write pa in place of you don't think that you will put h in place of x You are bringing h instead of x. If the value of one is found then the value of h is zero So here this is your limit h trends Two 0 instead of pi minus x h here pi and instead of pi minus x h now you know hey boss what happened to the limit sign whatever The angle tends to zero here p h so how much your answer has come your one has come no more Or you can do it like this, 1 is your pie limit tends to 0 sa apna h dosto its answer You do not get a direct answer, nor do you have any information I have got my pie is the limiting value of this function okay let's take a look at this It's a very good question, look at this will do in the question okay now look down your Zero is coming, okay so what should I do now? I have increased my limit to comfortably khana below x good from above x common aa It is coming to my mind from above x Common Aaha x common aya a p k x left below what left below Your B-in is saved, now be careful about one thing notice friend see i told you I told you brother x tends to 0 x tends to 0 sa x up x's answer is one or its Repo tomorrow limit x trend 0 x up sa x the answer to this is also very common a matter of sense hey okay so if you pay attention here x one over one over x one tens to 0 bit One, the angle here, that angle here, that angle tends to zero so boss what can you do limit x tens to 0 you have x up to x separated gave and limit x tends to 0 you have a I gave it to K x up B separately, what is the problem okay x up to x separate this separate so its The answer is one, so if it doesn't come then it is 1 multiply by here you get x to 0 You can directly keep the value of x as zero then K 0 K 0 now your friends have come up a pv up be direct this is the answer a pv up B okay let's go ahead a you this Look, it's a good question, it's good Question okay do it slowly what do you feel like doing I feel like changing the sign to no yes it is happening to me too so what did I write limit x tends to 0 t x i wrote as my cos minus san here here your sa why am i playing in cube limit sign Reason being see whatever angle is applied in the sign x she tends to zero so maybe that's mine This formula will work for you friends, right? Jaayega sa mine sa ka aka sa ka ka brother sahib ye what did you do, if you take LCM then it will be So the one below which is left is * If the sign is common from above then the limit limit x tends to 0 from above to common 1 - k I want to know why you are looking at me below If it is cut then what kind of squad will it become the same question now given limit x tends to 0 above 1 - below * 2s S look ahead, this is your limit x tends to 0 1 - sin2 to 1 - cos2 to 1 - c to 1 + c * 1 - c so I am doing it directly cos1 - c * 1 + c How to write sin2 1 - c s 1 - c s is written as 1 - c * 1 + c which is why the inter The minute form was coming, it was removed Now you friends can put the wall directly limit x tends to 0 1 up co in 1 p cos Direct if you replace x with g then it is 0 So this is yours and your cos 0 so this is 1 pw answer is your 1/2 gaya easy easy pg lemon q ji hai na a question You can do your Trigonometry homework from me. take it here is your Trigonometrical limits are also eliminated Just around this, you will get questions at this level. Limits will be asked from school exam Friends, this is a very important topic, very important Countless questions can be made on this topic And the difficulty level was also kept very high It can go but for now let's focus on are preparing for your school exams and doing a light competitive touch So I am preparing this lecture accordingly. I have come, we do it in other batches too good questions and more good questions, right? Friends, three-four more questions above the limits we will do it like this one, what are you saying, okay are the functions fx8 had placed the function above the limits fx8 now this is the same thing, what did he say to you that boss limit function find out x tends Now there are two ways on to 1, either left hand take out the limit take out the right hand limit look if both are coming equal then who will finish it The value means the limit exists and The answer will be your limiting value. The answer to the function will be starting now. I remembered it okay left hand limit You must remember the right hand limit a - a a p Then I also taught you how to keep it directly that brother can be a direct substitution Rashan egestion may happen You can have factorization or Using some formula from the standard limit If possible, you can use this thing in both ways. you can find the answer but what is the function oh friend, you don't even know the function f you know one it's not there look you don't even know f Something strange is given limit x tends to 1 function -2 is equal to -1 you get f one I don't even know, I put my mind to it what do you do if your brain doesn't function I know let me simplify this limit x Tendencies to friends can be given separately I can give you separately below limit x tent Up and down according to the algebra of limits I have given different limits di now this can go up to you why is that I am doing look at me limit extent tov f I want this I want this understood Now pay attention to what I did above limit extends to f A friends limit x tends to bhai its function minus function so its different limit its different Limit is applied divided by or equal to to pa multipla bye this is what you got here okay so it's yours limit x tends to 1 x s - 1 Now you know how to solve this Direct substitution brother x if you are one If you keep it in determinate form then it is not coming so this is your result x tens to 1 1 -1 Your 0 is pa*0 your 0 has come, did you understand this how come the answer row of this thing is coming with 1 instead of x square of 1 1 - 1 0 so 0 * pa 0 and this yours has arrived here limit x tends to 1 f x minus do you know the answer friends is the same as for any constant function The limiting value is the same constant as I got the answer limit x tends to 1 f off x -2 there + 2 this is what I wanted this is what I wanted so what was different in this question that You don't know the function, right? If you don't know, then its limiting I thought about how to find the value. that brother let me solve this, why not Algebra of limits is a very important The concept here is the algebra of limits In this you can calculate the numeration denominator in different ways. can give limit function minus function It is written its limit is different its limit is different You could have done the same thing here too. I took it there because I saw it this is happening here, this is what you are getting should okay let's do a very good one this is This is also an easy question, let's see what They are saying that you have to read the question carefully what are you saying yes gave the function this means If the value of x is less than 0 then the value of the function value if the value of x is 0 1 or 0 and one so the function is if the value of x is in between If your function is bigger than one then this is fine for what integers m and n do both limit Agitate well exists in both limits Limit exists means left hand The limit will be same as the right hand limit value So brother, I understand the limit It exists at point 0, let's see so first of all it was given to me that limit ex Tends to 0 f of exists does it mean that boss Limit Accumulates Tends to 0 negative f one is okay x tends to 0 negative f or else I know you can solve this directly also You can, but you have to do it using the first method Let me show you and it is fine with the 0 - h method Who will be equal to boss? This one will be equal to you. limit x tends to 0 Positive f is true for both limits the value of is equal now what will you do now Will you do this boss, I will first take you to this place I will do it with this mother only, right? Come on, this is yours. I have arrived limit i have limit h tends to zero because instead of one I am writing 0 minus A hmm 0 mine a good bye first write the function Let's say x tends to 0 from negative zero For the smaller one you would use a snap okay for greater than zero greater than zero For this you would use n a p If it is for something lighter than m then keep it The value here is limit as you have written 0 instead of x instead of x 0 - h times square p equals n Limit h tends to 0 0 p h p m okay now you know you can directly If you can put zero in place then what is the answer it has been coming from here yours has come from here Friends, if there is zero in place of h then this is yours and mine It will come to your mind, it will surely come to you He has come here and if his world cries then this If your m comes then n times m comes n times m Come n is equal to m, very good, now you Another thing is that the limit limit x tends to 1 fx1 negative will be equal to limit now one of the x tensor positive of function from one for smaller than just one for smaller than just For small n you will use one cup and from one of the elder for n ek p a use you will do it right okay so if you're here If you use this then your limit tends to 0 x replace 1 my a p a here you limit h tends to 0 remember starting what i did was our people put 1 in place of x h so instead of one you wrote 1 ps h's hole cube p m now you can open its hole cube and h Enter the value of zero or enter it directly If the answer is same then instead of h there will be zero friends or Your direct answer will come to you or your n will come to p a and here h will be replaced by zero This will also come as a direct result so in these Know what is going on with both of them You are getting to know that boss you are n and m brother you can keep any integral value of I have two equations in the form of n and m brother 2 = 2 3 = 3 4 = 4 right n and m are you Keep any value, I told you for what Integers m and n do both limit exist m and for what value of n are the values of m and n For which values does the limit exist? is at point zero and at point one m and n Why for all integral values of Even if you are able to solve both the equations You would have gone, I would have written m instead of n here what has come here instead of m2m n m2m m equals m it means that any integer value of boss n and m Keep it integral, it means it is an integer, let's go so what is the answer will take all integral values of m and n All integral values mean you can calculate the sum of m and n put anything in place m and n will be ans is ok Boss, okay, this is yours too. Let's go. Next again this is a very simple question But you people get scared seeing such a big question yes it is from NCERT so I thought I will tell you I will get it done from the front, it is very easy to be late a1 a2 dr dr dot a b fixed real numbers define a function function your given is the function given by look at x - a1 x - a2 dot dr dot x - a okay a1 a2 A what are your fixed real numbers it is constant okay what is the limit I asked you the limit. I asked you the limit. Friends, this was not a very easy question. you asked limit x tends to a1 function so what's the function your x - a1 x - a2 dot dot dot x - a I told you to find the limits Left hand limit taught in two ways take out right hand limit take out limit Did you tell me the direct method also brother? a1 you can directly throw it here 0 ba 0 The form will not be formed, if there is nothing underneath Friends this is your limit x tends to a1 you get it here a1 - a1 a1 - a1 a1 - a2 dot dot dot a1 my a1 - a1 what has come so that there is something in zero Multiply it too, and it will come out to be zero Limiting value of the function approaches zero for some good on point a1 and what did you say and What did the computer say? It also told you this. Find out the limiting value of the function very easy limiting value of function at x tends to a so very easy this is limit x Tend to a function What is your function Your x- a1 x - a2 dot dot dot x - a3 just put this directly here Direct Substitution Method I told you If the indeterminate form is not formed If yes then give your direct answer a - a1 a- a2 dot dot dot a- e3 you can solve this I can't even give this answer Direct is ok, easy, no problem, trouble no let's go based on your last question and Then I will tell you about derivatives I will take it okay, it's your limit Last question and this is my personal favorite question This is NCERT's very very good okay that's the question, well now you know one thing what would it be probably that just if the mode is written Your mode opens in two ways Mode Access Your x will open if your x is less than 0 or zero is bigger or your minus will open in x If you have less than one g then this thing will tell you You will know your mode in this way it opens to the one who doesn't know it boss then There is something wrong with the relation function, you know everything I have taught you, I will tell you again about mud x The value x will come if x is 0 or greater than 0 is the value of mode x - x will come if x is your is smaller than Ro, now someone will say Mode ki How did Value Minus come, brother, this is not minus yes this is positive how did i say If x is less than 0 it means it is negative If x is negative then replace it with a negative number If you put negative into negative positive It will become like this also means it is positive It's positive, it's just that, this is how to open it Would if inside the mode positive then it will open with positive mode If there is negative inside then you should understand that it is negative it will open from ok did you understand let's move ahead now what are you saying, I told you I gave you the function This is if x is a function less than 0 The value of the function if x is equal to 0 The value of the function if x is greater than 0 I can see one thing here brother remove the mode how will you look okay a is less than 0 if x is less than 0 So brother, the value of mode one if x is smaller than 0 So instead of mod one, my one is smaller than 0. mode instead of a - a smaller than a g so mode my you - a pv will come ok - a pv when Yours x0 than your x is smaller than 0 okay zero when your x g equals one if from zero If x is greater than zero then the answer of mod x is ax so mode you can write x or your x x when greater than zero is ok little Complicated but so much so If it is not complicated then first step It was clear, I just removed 'm' from the function Your x is less than zero means mode is one Instead of x you can write mine x your If it is greater than zero then write px instead of mod 1 can you now queen is very interesting speaking about what values of a Does it exist? You need to understand the question first. Try this for which value of a The limit of a function exists at which For what value of a for the value of c The limit of the function exists for Do you understand what I am saying brother One should be one in place, 3 ho -1 ho -2 be zero or the value of a can be anything yes no brother a then anything can happen a then Anything can happen I've No explanation, I have seen it in many places in the manner I tell you in the place from nowhere because the teacher himself doesn't Know what the truth is explaining I am telling you my love, I am telling the truth here But let me explain it to you very gently, first of all the limit You guys know the meaning of ageist here but one value I took the value of a as one value I took the value of a as one. If I say I'm thinking of the limit as x tends to 1 of the function value exists it means left Hand Limit at One and Right Hand Limit at Eight one will be the same means when I take the value of x I will put it from the right hand side towards one or If I put it from the left hand side towards one then It will give the same value, remember starting okay so just talking to you in one place What should I keep in one place or what should I keep in another place I shall give The limit of the function will exist The limit will exist i.e. right hand limit The left limit will be the same now friend, at the same place There can be many values, I have only one Divide the value into three categories and then I will check, do you know what the three categories are Anything can happen to you, but I have three put in category why can i put Look, either this will be your zero or so yours will be greater than zero either a your think about it being less than zero or a The value can be either negative or positive either zero absolutely correct absolutely correct understanding If I understood then something like this can happen yes or No, now I will check in all three cases Now I will check in all three cases. what kind of answer is going to come okay let's go So first of all let's do one thing on a e 0 Let's check okay that a equal to if row If it happens then does the limit exist or not How will they check, do you know your right hand? limit left hand limit so let's check let's do limit limit x tends to 0 positive fx8 to 0 Negative fx4 ro negative ok ro positive ro bigger than greater than zero which for greater than zero I will use it, it's a mouse, not a mouse If you cannot solve this right now then start from zero I will use this for elder one right Boss how are you doing it look comfortably limit x tends to zero positive f a I wrote a myve in place of given here limit x t 0 negative f one from zero instead of small print okay you guys know what to do now Here here friends you can directly limit h tense to 0 you say instead of one 0 mine A 0 mine A -1 here you limit h You say tends to 0 h tends to zero speaking yes ok h tend to 0 x place comfortably instead of x you wrote 0 p h ok here here there is 0 p h right and here is 0 - h here is 0 - h okay So this is clear to you friends how much did you get So instead of , you got -1 and instead of h you got 0 so this is your forest this is your forest very good it means left hand limit and right hand limit sorry left hand limit And if the right hand limit is not coming the same then what will you write that the limit of the limit is easy that the limit x Tends to X Tends to 0x tends to 0f x doesn't Limit does not exist when exists is zero The limit at zero does not exist, which means that A The value cannot be zero, what else can it be Now let's look further at lesson zero Let's see, okay, so now on to A Lesson Zero. what will happen just take it easy just watch it for a little while It may be complicated, but you should understand this lesson Zero means I am looking at the negative I am looking at any negative value. Any negative means -2 -5 any The limit on negative value will exist that will not do anything on negative value Will the limit exist or not, and then how? I will check it carefully give me limit comfortably limit x Tends to do a check f I do a check whether you are on negative a y If you want to follow your lesson then brother what Check on Limit X tend to a negative A negative means y from left hand side I'm going towards that value, brother. some number like this will be mine in the lane human would be okay suppose the value of a human means that it is human negative I am going from the left hand side towards -1 am and and and and and and and limit x tends to a positive what would that mean limit x Tends to a positive means here Imagine -1 written positively -1 Positive means you are on the right hand side -1 can you imagine what we are heading towards you are keeping a lesson in your mind 0 have to assume function and here also function okay now relax okay a negative means There is also a minus, right? So minus before minus before means the value of x is also less than minus Little friends, I will use this - x + 1 so Friends what is your limit here x tends to x tends to a negative - x+ 1 And here also you brother this is slightly less than -1 You are considering it to be big - even slightly bigger than 1 It must be negative, just slightly bigger than Mive Even if it is negative, I will take this only - X pw what are you understanding yes i understand Guruji my one pv is coming is this Do you understand what I am doing, this is not it Think about whether the value of one is greater than A. hey your negative number is negative number I will add a little bit and that too will be negative will be very slightly negative meaning of x If you want to use negative input then solve it what the same thing The limit h tends to 0 at x instead of a − h + 1 And here the limit is h tends to 0 h tends to 0 x instead of x instead of a + h + 1 ok, solve it if you replace h with rho If you keep it then friends this will be your minus of a + 1 left hand limit if you want to find the value of h place zero so here also your minus a p will come now what happened here left hand Its limit is coming same as right hand limit What do you mean by that you say limit extends to A When Your A Lesson 0 is a function of will do it, did you understand this means now here brother what did he say which of the a What are the values of a for the values If limit exists for only one three The values of method can be either zero Either on the negative or on the positive row I left hand right hand on checked row Limit is not coming same i.e. function on row the limit of will not exist means a the value can't be row so I typed a the value less than 0 to the value less than a Left hand limit Right hand limit ranging from 0 You took out both of them and got the beans, meaning Meaning means meaning one is less than zero Function key for values less than zero The limit will exist i.e. the value of one All negative numbers can be the same The values can all be negative numbers I asked you what values are possible If yes then all the negative numbers of A are the answer this is your homework or you Let me know in the comment box This is your homework or you can tell me in the comment box I will answer it friends, I will answer it Brother, I want an answer in the comment box It is found that the value of a is not zero The value of a can be any negative number This can happen for all negative numbers The limit of the function will exist positive For the number, you tell the limit of the function Exist will not exit in this manner you have to ok let's go with this friends The limits part ends here Isn't it true? I hope you understood the fun too. must have also come now why did you read its limits There is a region behind, this is the beginning of calculus Now we will study derivatives and I will teach derivatives in this way next time Please give me the next 20 to 22 minutes Solve questions on derivatives Many people don't know what is differentiable I wouldn't have known you this way I will tell you that you have not read anything like this anywhere Let's start with derivatives Now let's talk about final derivatives So first tell me one thing, do you remember the meaning of slope Derivative is nothing, it is slope, I will tell you how is it ok now let's say i have coordinates axis li okay let's say I have coordinate Axis Lee now I am making two lines for you I will tell you in very basic terms. I made this line in front of you and I You made this line in front of me, tell me these two Which of the lines has the steepest steepness yes steepness means slope okay Here I write that slope means nothing happens steepness happens steepness I mean, you can think of something like this you can think that Where will the boss climb more? You will climb more Brother, I feel like this is a way out, this is also a way out Brother, there is a way for you to go from here to there You don't have to work hard this is steep this is more steep than this yes or no brother its slope will be more than this do you understand what i am saying do you understand ho what am I telling you slope I taught that the meaning of slope is tan3 m = tan3 y2 - y1 x2 - x1 You will remember but its what does it actually mean learn that brother both are lines okay its slope its Slope will be something or the other, slope means Where do you find the steepness of the line the most? You will have to climb higher here Will have to climb, friends, this line The steepness of the slope will be more understandable First common sense is ok, now friend like this So there will be a lot of lines, lines of lines In what way is the slope of a given number assigned a number? I will teach you that too brother, which was given by Do you know how to find the slope of a line? How did the formula come about and where did it go I will tell you everything about how to reach there but for now First learn the common sense meaning of slope Of these two lines, why is this more Because boss its steepness is too much Can you understand the meaning of steepness as climbing? You won't have to climb much here I understand what you're saying ok Dude, you are telling this after seeing it randomly Which is more, which is less, whose slope whose slope is more or less if I measure the slope Let me give you a number, meaning if the slope of this line I will give a number to this and similarly to this If I give a number to the slope of the line also It will be fun if I give it to you, how was this number given I will tell you now but before that I have one more question Tell me brother, you are from here to here go here or go from here You will have to work hard, think about it I'm just telling you that, I mean you Suppose I bring random one here Will you go from here to here or from here to here You will feel the same slope as you go Will you feel the same kind of surge? Yes or No means the slope of the line is constant through out did you understand this line The slope of the line at every point is equal to If I talk about each point then each The slope of the line at the point is constant How to give the slope of this line as a number gone let's talk about that okay one thing is clear that boss slope of a line is constant ok now how did you give this number okay friends pay attention I took these two points and one point here I took one point okay I'm on the slope I want to take it out okay you just need to do this Let's see how far you went from this point And then we call it how much it went to the side You call it rise, you call it up run yes rise this is called up run me Tell me, everyone knows how much this distance is that this distance is yours then the rise is yours You have come here, this distance is your one, I can see it It must be brother this is your forest so run is your forest I went to friends Rise over run is called slope, rise to of run one your slope of this line to a then give a number to the slope of this line diya and you can check it at any point assume that you assume that you understand Friends, I took this point, this one and this Brother, it takes a long time to go from here to there How much is the over run rise? Count the rise amount count it boss this is yours 1 2 3 4 5 6 six rise run how much is 1 2 3 so 6 / 3 dekho to The slope is here, do you understand what I am saying? Take any two points on the line and rise If you look at over runs, it will always come equal to two why because the slope of the line through out every point is same It is clear, it is clear now friends pay attention Please give me what I taught you to hurt me Boss, I have to find the slope of any line So brother look at its inclination angle theta and If you don't take that, then theta is your slope. if it was equal then what was this? It was a useless thing, it is the same thing, it is here I did not see anything friends just believe this I understand every small thing about this angle to obey theta said okay well if One who cannot read straight line should understand this If you are not able to come then what is going on then please tell me Ignore the minute I will come to the derivative but this It is important to explain, okay, man, what was it So it is obvious that this is also theta will obviously be theta it will be obvious the common cusp angle is okay Friends, just look at this triangle Just look at this triangle carefully So if I take the value of this theta then it's not theta what is perpendicular upon base How much is the perpendicular to base equal to how much? It is equal to one, rise over run Nothing is written, only rise over run is written If yes then understand the slope formula Why do we use tangent in inclination angle that tenge is nothing that tenge of theta to you Nothing, Rise is giving over runs only He is giving over runs, I understood, I understood It is clear that there was another way, slope Take out what is important for you Shall I tell you that you know the line To find the slope, you can put any two Collect points as if you have taken these points what are its ordinates -1 1 -1 -1 sorry And it seems like you took this point. what are ordinates 1 1 so you know pay attention above the line pay attention above the line Points are -1 -1 and one is 1 1 with this you can also how can you find the slope remember y2 - y1 is divided by x2 - x1 so y is divided by this brother you can call it x1 y1 or you can call it x2 y2 you know y2 - You can also find the slope from y1 up x2 - x1 yes this is very important boss so what y2 how much and formula's minus y1 your -1 okay is x2 how much is 1 minus x2 of the formula your -1 Friends, this is yours 1 + 1 2 Did I do something wrong -1 -1 and this 1 1 Did I make some mistake? Why do I feel that I have made some mistake?1 , 1 I made some mistake, just one second Its coordinates are -1 -1 ok yes friend this was a mistake don't understand I talked about its coordinates, hey friend, what am I doing with this Coordinates would be living beings, just think about that How can this mistake happen brother, see the slope is coming apart while the slope already knew where to come I should come, I already know it Slope two should come, right friends? Coordinates brother this line is close to Nav I am not doing it no, I am passing it with all my heart Thinking sorry, she is passing by with love, I am the same If you think something is wrong then it is your fault It's done now, just apply the formula and Is the formula the same, boss, do it comfortably So look at y2 - y1 or your y2 or your y1 You can write this if you want but you can understand it So this is 1 minus -1, then this is your 0. minus of my friends your 2 up 1 look When slope two arrives then you can do it by any means Calculate rise over run using Tange's formula or by the formula y2 - y1 / x2 - The slope of x1 is the same as that of your line It is clear that it will come, it is clear now the thing that is important for you is that the slope of a How did you find the line and the slope of the line The through out remains the same but now the question arises but now the question comes but now the question comes comes hey before the question before the question let's go I'll tell you this before the question Friends, Dale Y, I just told you y2 - y1 now up x2 my xv y your slope what happens is this is your slope y2 mine wow to Basically I can call it change inva hoon change inva and x2 my xv basically you can you call it change in x change in can you say x if you understand then slope it gives you nothing basically what How much change is happening with respect How much change is happening in two x How much change is happening with respect how much change is happening in to x change inva I change it to x and I write it like this Lava up del one just listen to everything Everything will be clear in the next 15 minutes. what am i trying to say lava up del one ok is asleep up Upon Very small very small change in one is ok very much Small change in wa and very small change x I'm in a very small change and I'm in a very small okay in change x now the question arises that just the slope of the line so the slope of the line so every it is same on point ok but how do you Find the Slope One the Slope is Constantly changing if i ask you brother here The slope will be different, look at it like this from here The slope will be different going here from here There's a slope going here now it seems like this is the way If you believe this is a mountain then go from here to here You will feel a different steepness from here You will feel a different steepness while going here Basically brother this curve or whatever Function The slope of this curve changes at every point it's happening na so how do i find the slope When the slope is constantly changing when the slope is continuously changing when the slope If it is changing continuously then I will change the slope How do I find it out, I will tell you I believe you are at a particular point slope suppose you find the slope at this point Have to find out the slope at this point To do slope finded out I mean you are finding out the steepness nothing can be done at this point Basically a line is drawn from this point You know what you call tange, Friends, you call line as tange Friends, this angle will have only one slope slope of tent listen slope of this The tange slope of this tange is called derivative slope is nothing derivative yes I was telling you about this tension again and again Brother if you need to find the slope at this point if yes then you can pass a tenge at this The slope of the tange eight will be yours The derivative will be the derivative at this point This point means the slope at this point what did i say what did i say derivative at this point just keep listening to it things will be clear The derivative at this point or any other point could mean slope at this point so if you are taking the derivative, that means You are taking out the slope and don't understand anything Come note this first you can use derivative If you are taking it out, it means you are taking out the slope of such a curve or a curve of such a curve At the point where its slope is continuously If you understand that change is happening then derivative at this point is called slope at this point The derivative at this point is called slope at This point is clear, but find this slope how can i remove this slope boss How to find out the slope of a point I made the tent but how to make its slope Let me find out the formula carefully Rivescript Tenacious on Every Point The slope is changing well by the way Let me tell you another thing: derivative at this point Tell me the slope at this point or the rate of change tell me what do you say rate of change is Why Rate of Change at Points Rate of Change I am saying this because what does slope mean is change in y my change in x change in y my Change in x means rate of change of y with respect to x rate of change of y with respect to x so at this point I Basically I am just finding out the rate of change You will understand it in just a little while See, that's why a lot of people rate derivatives off Let's also say change, okay, at that point. I am finding the slope, that is the rate at that point I am taking out the change at that point I am taking out the derivative, everything is clear give me 10 more minutes, that's fine Friends, you have a curve, as if you have to go to this point but you want to find the slope, as if you have to find the slope at this point But I want to find the slope, suppose I took this x I took this Friends, I assume that I took this point as a so what is this will f a will happen that's right this will be f there will be any problem if this point is crossed If I took a then assume it is your f f okay okay function your y = How to find out slope on fxxx.pro do you know the formula that if brother do you know how to calculate tangent on tangent if If you get two points then you find their slope can you out how same y2 - y1 up x2 - x1 I just had a few thoughts far away a little bit far away it's like here Distance But there will be a point at a distance h very far the value is small h very small value is very Small value is ok listen h distance assume a p a will happen obviously brother maan if This would happen if the distance was 2 x 5 okay so friends what will happen its correspondingly I understood what I was saying, I understood what am I saying okay now two things first The question is what are the coordinates of this point will be a f a a f a correct point what will be the x coordinate a + h y coordinate f a + h do I understand this I am telling you, look at it carefully and don't get confused Every little thing must be understood So can you explain to someone else what I am talking about? I'm interested I'm on the slope at this point am interested in taking it out or else this To find the rate of change at a point I am interested and the slope of this point At this point I call the rate of change I say derivative at this point I say derivative at this point At this point I say derivative what am I doing I'm taking out the slope slope at point a or or or I'll call this I will say slope means derivative so derivative at point a or rate of Change at Point A Wow how fast is it changing How fast is comparison to ex changing? have understanding gone okay it's clear boss okay now concentrate See how I get this out, friends tent or else you pass a second right now It will be hot brother, I will pass with two points which means I passed you a line can you give me this Can you tell me the slope of the line easily? The slope of this line can be told on the line you know the two points you know the slope Consider it the formula of I called it p point If I say q to this then you know that what is the minus x coordinate look what If A is subtracted from A then this is your slope of p. the line has come pk line or second line has come f of a p a my f a up h watch it comfortably but what i want the slope of p k no no no i want p do I need the slope at the point p I want a slope no no no at the point me at p point me at p point slope Friends, if I want the slope at point p if i want i can say that yes brother it is still some distance away if this point if this point is related to this point The distance would have been very less, reduce it immediately reduce it completely, reduce it completely so this q Think that point p will come above point Look at what I'm saying, think about it what am i saying bro if this h is to be done right now you have some h, suppose it's your h5 right now will be If you reduce it, you will get the answer for better slope I can take a deep breath again comfortably Let me show you another way, I understand I will not be able to understand this way again so if this point happens here understand that if this here is the point I would like to know what is the point in this but what will be the slope i have taken from this point I took a point at some distance and If the slope is taken out then it is somewhere near the slope of the pack The slope of your P will be around the same but if it what is point and pass what is point and pass if it were so then its slope would have been closer would be the slope of pi if it is a point and as if it were here as if it were here So the slope at this point would be quite I would be close to the slope of this point I am interested to know that at this point What will be the slope of the slope of this point If you understand what will happen then I will just what i am trying to say is if this q point p point comes to comes to comes to if p the slope of the point or the slope of p I'll say it again if this q point p coming closer to the point coming closer coming closer if p goes on coming then the slope of p will be will come very close to the slope of and I will get to the p if I want to find the slope of a function then should I say that I can say that brother is coming to q p means h it is decreasing na q it is coming closer to p it is decreasing it means first it was five then it was four imagine then your h you imagine h3 then your h you hey brother it should be written nicely h your th mage then h your two then h your one then h your something tell me the value of h what happens to the value of h such that p The slope of is the best approximation I'm interested in knowing the slope of p yes you guys have answered, did you understand what am I trying to say okay We have to find the slope of the point q which comes very close to If we go then the slope of the pack will be brother maan maan Right now the slope of the lake is coming near it it would be ok but this would be the slope of the it will be closer to it q if it comes this way then it will the slope of is closer to q right next to it If you come right next to me, then the slope of the Pak and will come to me on the slope of the P I am interested in what happens if the slope of p is may I get it What will happen to you friends, yours is zero go mean understand mean understand if I go to this The limit in cheese is that it tends to 0, which means it is zero is going to what is the value of h the value of h is approaching zero The value of h is approaching zero. is approaching if the value of h is approaching zero If it is going close then the value of this slope Slope Key The value will come if you understand then limit is trends Two 0 of this slope The value of this slope will be the slope off point p will be the slope of point p I understand what am I saying understand what I am saying it is clear what i am saying to me I wanted to find the slope at point p. have you done the limiting value of this The limiting value of this will be the slope at point p limit value of this n x tends two zero will be the slope at point p What does limiting value mean when x is zero? if it is going to then where will its value go when h is approaching zero then Where is the value of slope going when h is zero is approaching then where is the value of slope I am going to take this out so I put a limit and this is what I call it is this what you call it what is this called derivative derivative derivative at point a what We will say that the derivative point is Bhay point P But I am taking it out on x, right? Finding the derivative point, understood it My friends this is how we write derivatives See how to write derivatives at points Means slope at point Derivative at point a means slope at point A you can also call it your ad A and its what is the formula limit h tend to g f of a p A–F hey you understand this is for you it is important it is important for you It is clear, I hope you understood it Friends, I will tell you the meaning of derivatives I hope you were successful in explaining please Let me know what you think about Ed What is the meaning of derivative at point a f ex derivative at point x well by the way this If I want to change x to another simple Which formula you will use, if I I have to change it to x okay suppose if I f I'm talking about f general point right now I've got a Let's talk on point, now any point can be made so if I wrote x instead of a If you write x in place of a here also then The limit h tends to 0 at a instead of x + h a place x up h ok do you understand what this means derivative at point a sorry point acus derivative at point x or then slope at point x derivative at point Ace or slope at point axe is fine Friends, this is called derivative remove this by the first principle It is called the formula of the first principle Taking out the derivative by the first Principles by the First Principal, okay, okay, it's clear now. let me teach you how to use it once Let me teach you these direct questions Learn it, it will be fun, it will be fun, okay now believe it I saw some function, its actual function is not working I will tell you ok by the way tell me one thing You must have memorized the formula, what did you memorize the formula would be bus bus bus bus bus bus if I I will give you a function, a definition of this what is derivative 2 why does the derivative of ex come to 2 a why does the reason come behind it he is telling me take it easy, okay come let's see Let's take an example, it will be a lot of fun brother Now this extra thing is more interesting I am telling you so those who don't want to read it have nothing to do with it no you can skip 10 minutes ahead and go to first you can see the principal's questions okay y coordinate x coordinate y coordinate ax coordinate very Well, friends, I took a function what did I take of x s y = x s function taken or you can also call it fx2 you can okay okay suppose I'm here Finding the derivative at a point Meaning we have to find the rate of change at that point Or you need to find the slope at that point all three things are the same as if I have I took point three as if I had not point taken I took three, okay, rate me on three. I have to find out the change, find out the slope on three Do you understand what I have to do? But you have to find the slope, did you understand this? It's okay, you haven't gone, just pay attention, it's okay So the derivative of f is 3 or the slope of the path what does it mean to take it out okay friends If the value of x 3 so fx9 so i can say this so i can say this Now I know this nine will come I need two points brother one I will not be able to find the point slope of two points so second point i guess i have slightly h distance p li 3+ h 3 p then if the sum of x If the value is 3+h then tell me what is the value of y If x is replaced by 3+ h then the value of y will be 3+ h Of Write down simple square coordinates quickly So what will be its coordinates? coordinates your friends got 3 9 and what happened to this x coordinate 3+ ava Given a hole square of coordinate 3 p h Just understand what I am doing I will extract the derivatives in the path I'm interested okay so so so so so so so so take out the slope okay on Derivative m at p 3 Slope at path you will know what will happen what will I do, what will I do, just brother no If you can draw that line then draw the line What will be the slope of this line or minus wa so when you minus wa when you you will minus when you or you tell him to do one thing okay The slope of a line segment or line Will be 3 p h's whole square minus 9 ap 3+ a - 9 3 p h -3 Sorry -3 went from 3 to 3 so the slope came at p Your 3 p a square - 9 up h now you have you know you want to find the slope of p so this You should become absolutely zero then should I I can say slope at point p slope at Point p will be equal to limiting value Off limit x tends to 0 limiting value of do the same thing 3 + h s - 9 / h solve limit I taught you how to solve it will do boss in determinate form is first We will convert it into determinate form If it is normal then this is your 9 + h s - 6h sorry + 6h -9 / h is correct 9 to 9 went very good So here you go limit h tends to 0 h s + 6h upon a from above h common then h p 6 upon a a to get the value of a with zero slope your s a what does it mean to listen to derivative at path Yours has gone derivatives at Point 3 or the slope at p for rate of change at path your 6 has arrived understand it, it is clear it is clear and this You could have come directly too, know how Let me tell you about the direct six, come see that too here look at that too ok i have done this again How does direct six come, all this panchayat There is no need to do direct si How do I come to what I do After you get things cleared on 100 I will ask questions after looking at the formulas, okay? Friends, that is your graph of f equals s it seems okay now you understand on p3 took out the slope then it will come to p it will come to p8 There will be different slopes at different points why that was the intention that if the slope If the slope is changing continuously then How can I extract different values at different points? Now if you use three then slope on three A six came as if it was a point 8 then it would have hit that The slope would have been different if the point was 16 then The slope comes separately, now brother this is long I will think of a method to remove some slope Putting f = x Friends, consider this point I took x So if it is x then if x is written in place of x then fx2 will become fx1, it's simple, ok Suppose here I have placed a point x + h at a distance x + h If we take h then x + h corresponds to what is the same thing quickly its coordinates what will happen x x s ok and what about this will become x + h = x = h squa understood clearly brother same thing line drawn line Let's pull it a little bit, right? and very good ok bhabhi accept this I told him q then slope at slope off What will be the line p of y of x + h square - what is the square up of x is x+ h - x y2 - y1 is doing the same as x2 - x1 No, now I know, it just cut me I better I want the slope on p so on p To find the slope, you know you need to Its limiting value will have to be taken when your H is tends two then x + h S - x S / H Solve this quickly, then limiting value off h tend to 0 a spa a spa 2 x If my ex goes to one of my places then this Your limiting value of h tends to 0 a common h p 2 ok ok friends i am going to aa h Look at the zero place, 2 one has come, 2 one has come, meaning but the slope of your 2 axis is coming to point x But slow you understand derivatives rest From derivatives at Point x equals to 2 one aa I understood I understood okay and that is why you must have remembered the formula that a square has a direct derivative There is 2 and this is where the first principle comes from direct derivative of ex squared Yours is 2x, now you understand that you can earn Rs. 2x at any We have to find the slope at the point, as if at th I had taken it out on my wrist but the slope was yours Six was coming brother, put th in place of x 2 3 S will come as if you were to climb on a slope F If you wanted to take it out, what would have happened, just put it here Slope 10 If you understand what I am saying then you You can calculate this function with this simple formula For or else for this graph you can use any We can find the slope of a point, if If I ask you at this point, then at this point the x coordinate of will be negative the value will be -5 So the slope will be -5 -10 Understand what I am saying, it's clear So friends this is the funda of your derivative derivative is nothing slope is okay And I told you where this formula is from. I taught you everything, now you know something have to ask questions Now you have to ask some questions, okay? Look, you can see the formula written here too. This is the formula for derivative at point x and the derivative at point x you just remember put the derivative at point x i equals two Limit h tends to 0 fx8- fx8 friends this is from first principle The formula for finding the derivative is Let's see many questions based on this Now you are asked only Question Question ok first of all help the first principal taking the derivative from and then I I will give you the formulas directly. I said derivative take it out, okay two Write it down as per the first principle here. What should I take out as the first principle? it is much easier than it is much easier than it is ok so what should we do first Write down the formula so you know what f' is x or else do one thing first let's write it down I want to find the formula f x what is the formula Limit h tends to 0 f a p h my f a up h okay its clear now listen carefully my opinion Friends, first of all see, you know f yes f one p h in every question you just f one p h take it out, how to take it out is simple Brother, put x in place of x h hey put x in place of x I don't understand this, it confuses me what what what what what instead of x+ h x in place of x+ h so here in place of x we have x+ h so 10 * x p h understood just put the answer here will reach the limit h tens to 0 f x p h of place 10x p 10h f one place 10x up h 10xx So friends this is your direct link Limit h tends to 0 10 up h h to h went your answer is 10 your answer is 10 ok easy and you Friends should also know that this is a It is a straight line, right? If you look at its graph If you make a straight line, a straight line will appear The slope is the same everywhere Your slope will come 10 everywhere, every where The slope of the line will not change next Look, I said derivative, look at it first What should I do, I have to talk to the principal only I am doing it from first principle right now what did i say first of all If you write the formula then the limit sorry f d one well by the way you can also say this to f d x ho d vap h tends to 0 f of x + h - f x up h most You already know f x from looking at the first formula f x + h always precedes f x+ h x instead of x + h then wherever there is x there is If you want to write x+ h then 1 up x+ h will come I understood it, put the limit in two formulas and solve it If you know how to do it then this is your limit h tends to 0 f a p h means 1 its x+ h f instead of one you wrote 1 up x up h you Try anything, you can directly replace h Cannot keep zero because indeterminate The form is coming, I have taught you this already i limit h tends to 0 from lcm If you take the LCM then friends your result will be x p h in x above your x over here your mine in x + h hole divided by h write one below h If there is nothing then put it in, then this is yours limit h tens to 0 x - x - h now this gives you comes so I'm speeding up faster and It's not too fast down there Friends your result will be x + h in x these together I got multiplied on top of this and this okay among themselves now see if x is subtracted from x then This is your limit, tension 0 - h up x + h * x h okay because Indeterminate form was coming, he was removed If you understood then it is yours Limit h tends to 0 -1 / x + h * x you can put direct row instead of h If you can write it then instead of h you can write it directly in row -1 0 friends your x s will come down So this is your derivative of this up x Derivatives of Mine Up Ex It is clear for sure, some more questions first ask three or four questions to the principal and After that direct formula based questions let's do it very easy very good let's do it Let's go to the next step friends and talk Let's start with the first principle of the bus Derivatives or what difference will there be ok let's go friend c x You guys might have memorized it Formula-like x is how it comes Look quickly once, what are you saying what is the same thing so first of all you know that f one here f one is your k x tell me f2 what will happen oh it's simple k off 2 f3 what It will be simple, just apply cos3 like that no brain f of x + h what will happen to x put x + h in place of x and anything else no okay let's do it so you will get the formula happens to f'x you out is f d a by the way fxxx.pro h solve then limit h tends to 0 f x cosx-sinx + h = cosx-sinx minus cy a sa b minus k x our h now here guys I am telling you one more very important thing I'm going to tell you the derivative fuss principle Work on trigonometry questions What did I say that it will come to take out the derivative Using First Principle in I did well in trigonometry questions I told you a formula, what is it x up x limit x tens to 0 and this happens to you Now remember one more thing along with this. very important thing okay boss if The limit extends to 0 is 1 minus x up to x it is written or or or k x my up x is written Its answer is zero, remember this very important How does important come, hey that same friend x change it to sa one and then this The answer will be zero, remember this please ok what did i say about limit Whatever the angle, it should tend to zero 1 - k x or k x - 1 is up x, its answer is Friends, your result always becomes zero, this is the thing How will you be used here if you pay attention If you notice, I will give you h below this also even under this even under this if I am his brother because of this h Indeterminate form is being formed if this h is removed It would be fun if I go under it if I give it then it becomes h up h10 0 a up h h tends to 0 and happens to both of these What to do between these two friends if you Take out the common and see what happens if you use these In both cases the limit h tends to 0 if you calculate x If you take out the common number then you will be left with x and -1 Put h below it and here is your - put sa x in sa up now you know its I have told you the answer, I have told you the answer Told you I understood clearly what I am saying okay so limit t-up friends 0 so 0 in will yours reach zero, okay limit is tend to 0 sa up will come to mine and that is why a Derivatives or The difference would be mine yes it's ok it's clear it's very easy let's go Next let's do, what did he say to you guys asked you guys a question about the derivative of t take out x it is very easy again same thing f is given f is given find f of exponent a then f a p a will be n off n off a p a it's right it's right do it boss so the formula you know is limit h tense to 0 you have to find f ex limit h Tends to 0 f a p h - f i am ok very good so this is your instead of go limit at f expa you have you wrote off x p a f in place of one I wrote no of mine now it doesn't matter brother please change the sign to cos then Limit h tend to 0 or a partner like you h your one p h minus here your one And here's your d x and below that's your LCM Otherwise, I would feel like taking it brother in termite The form does not get deleted in such a way that the LCM is If you take it then your case is of one p h in k x it will come okay there is h under it that's right Below this is your h, under the h you put v So that there is no confusion, what is on this I went here and yours arrived of x + h I multiply x by you understand went to minus sinx-cosx First simplify it, it is not good, see it directly Don't be afraid, it's on top and it's between us now look at this x p h k x p k a sa Do you have an identity I am not able to see what b minus what a side B I can see this is happening to you friends sign limit h tense to 0 sa a my so ye yours x+ h x+ h - x and what comes below will go down your friends has come k off x + h in k x * h is the root of the indeterminate form Friends this is h, from x it went directly to x Let me do a little bit, it's up, it's up, it's limit h tends to 0 so h is up to h1 then Friends, I wrote this directly. And here the limit is 0 what is your savings 1 / cosx-sinx Let's clear it, let's see what he is saying now I will do either one of them, let's do one Let's work my friend, it's easy my dear so let's do this one okay first tell the principal, come on do it quickly friend what are you saying the formula is the same or not Friends I am doing A off here let's go or given to you f one so f one given to you x pv your x myv f x p a what will be the replacement of one x p Substitute x p h -1 in the formula instead of p 1 directly keep the limit h tens to 0 what is the answer firstly f p h then this will come x p h pv up x p h-1 minus f comes from f p h - f formula I am not writing, you will understand that it is so long If this is a lecture then you must have understood it by now p what f toy your friends x p 1 up x and What is going to happen below you? This is the root of your troubles. now it will take LCM from LCM The answer will come very easily and very quickly, so the limit h tends to 0 x p h-1 in x-1 what will come up yours by going into this This multiplication is x p h pw in x-1 here x p in x p h-1 okay and friends I am here to tell you this I'll multiply it a little directly I am doing it so that we can save some time after this brother let's see then brother open it I will have to say, I would have opened up a little bit so that you feel that your teacher too He is working hard for you, isn't he x go x go x me then x s x going to minus minus x I am your friend h a go to my place I so one goes to the x then this is your x don't go myve myve look here now rest army minus x going into x then minus x s minus x going into a then minus x minus x goes to my side so p x understood then go to my x I am a brother sahib I am going to be a isn't it and if you go to minus mouse then you will get pv Saab it's been a long time coming, no problem friend It's okay friend, this much is below x p h myve in x-1 these right now look up something is getting cut what hey a lot of things are being cut look at one s to - a s my ex to p a p a to - a p h - h a okay up top your -1 + 1 above only your - h - h so - h - h is - 2h so here it is for you Limit h tends to 0 here you go minus 2h below your x +h -1 x - 1 * h Friends, h was cut from h which It was the root of the problem and has disappeared now that you are If you keep the place directly then it will be yours for -2 h place 0 then x-1 * x-1 of x-1 squa this is your answer got -2 up x - 1 square easy this is yours homework this is your homework is clear come on ok friends these two are also on the same pattern You guys please do your homework I have to tell you about the first principle I showed this also by doing two, three or four Extremely Extremely Easy Questions Hai Finding the derivative from the first principle okay let me do this one now one more After this I will give you a direct formless I will tell you with the help of formless you know Its direct answer is know how to use formula From or your direct you can write this ok but we were told by the first principal If the derivative is given then first you have to find it from the principal Friend, this is easier than what I can do come on, the second one told you I said f gave you x That's right, your homework will be very easy. If this happens then this is a very fu question okay you tell me what is the f of ex will be f p will be b just x instead of one instead of px use p to sa P h I understood it, okay, you know the formula Friends f One limit h tends to 0 what was the first f x + h then instead of f + h should I write this directly should I multiply it by x sa x+ h p h in [music] Sa x+ h is the value of your f x+ h - f - f x e xsx4 I will do it, it is a pain friends yours has arrived limit h tends to 0 x * sins.com ok friends x was outside instead i did this and this is written meaning something like this It will come okay, there is no problem no problem ok what happened here plus here also same h sa x p h ke place sin a cos instead of sa exp a b plus cos so B Bracket Minus Saw Hole Divided bye i understood the whole divide ba what should I do now what should I do now Now pay a little attention directly I will do these two when I go X If I multiply it, the result will be 11. sa a x a if i take a common Let me see what benefit I am getting If I take out x from x then x will become x if i went then left and here x is equal to -1 left and after this I have put it separately below, this is it this also has it, this also has it, now these friends x c sa easily x c sa x x sa you understand it okay so let me write it down but friend let's do one thing, I will open it I'll take it, what will it take to take one more step so x sa a k h ps a k a sa very dirty You guys get confused inside h da h sa a ka a a a a ja inside h ka a sa here But yours x s one is ok ok boss let's hole divided by your h Now pay attention to what I am doing Out of the two, did you understand these two? should I make it red from this red one you If we take out one common x, what will be the benefit You can know the answer to the question directly if it is -1 up Now is there anything else that looks common? do you see anything else common x k x Something common is coming from both of them It is visible from both of them but then what about this you will be fine okay if i have something in common with both of them If I take out x cos a pa x cos Ex plus h cos acus Something will be left, something is going wrong with me Something is wrong, something is wrong Mess Either do one thing Either let's do one thing, let's see it like this understand if I have something in common with this one and that one If I take it out then there is a x k x p h inside It will be alright but then what will happen to it here but then you will be cut off see this is your a set it will be cut off, okay it's right, you are absolutely right in these Find the common thing from both, this is where you have to think everything has to be thought about here, okay something like this It is not like I learn it by heart like this you would ask the question this doesn't happen here let's think about it all, let's limit it to zero or your done one cy one kisa a or Then if we take direct common then x sa ek what is left inside the common A from here I gave my A down What is common in this and this? If two saa common may be visible then what is inside left a k x a k x and here you have the rest ok or else you can do one more thing x also take common na sa ka x sa ka x ha I couldn't see it at the time, common is left here x is left here h ours h ok then write down yours h sa one by one you now see the whole root of the trouble The limit of this thing is being cut h tends to 0 value 0 then 0 * x sa x friends your 0 ia Will Go Plus sinx-cosx will your plus come here Limit h tends to 0 sinx-cosx h0 then k01 this is yours If you apply sinx-cosx formula you will get the same answer will go immediately okay so now this is your homework this is the one, I gave it to you as homework I have given it to you, please keep it, ok bye Have you not forgotten The Algebra of Limits? Brother, this is the limit of this one as well, this is the limit of this one as well This is the limit of this as well and this is the limit when you are in this If you put a limit on this, boss, and this too limit its also algebra of limits is ok So here from your first principle The questions are now finally over The last part of the chapter that is formless now I am giving you the direct formula. what is the meaning of direct formula brother if let's say your function f x sorry f x your x key is the power n of its direct derivative Now you can find the formula here first Need to take the derivative from the principal There is no direct formula what is n raise x to the power n - 1 f ' x look at this is exactly the derivative of e to the power of x e to the power of x means if power x then f x is your e to the power x will go or you can say it like this brother d bye ki boss d bye is this yes d ba d one of e to the power x your e to the power x would be you can also say it like this, okay a to the power of x Graph of the exponential function of a power of a friends of x is a to the power x l a where p is your law of x greater than zero. These will be useful to you, see these two will be useful It won't be that useful for now it's ok l x it happens 1 / x okay remember all these you must be absolutely memorized, meaning if fxxx.pro and can do that for the time being it's okay sinx-cosx it happens, remember and sec1 x is ok cc-1 happens - cosec2x and friends The derivative of any constant is your zero The derivative of a constant means that if f If I gave you f ex then yours would be zero Simple derivative of any constant Yours is zero, did you understand, ask questions I have to remember everything, so let's go, number one question ok by the way now question Before I go, a few things came to my mind you will have to learn from let's learn or Then let me ask you one more question here, let's see This is the question This one hey look at this first friend remove it When we are finishing, everyone finishes it, okay Now just as the algebra of limits was the same Algebra of Limits Your Algebra of Does it work in derivatives too, boss? If the derivative of the addition of two functions is written as then you can find the derivative of both the functions You can also add derivatives by taking different of two functions are all written then you By taking the derivatives of both the functions separately you can also subcut it ok but what happens in multiplication and division I will teach you for that very easily right now. What is multiplication and division It happens first you learn and then I ask questions yes, let me give you an example as well, okay what does the product rule say if it is as if given function let's take it like this let's assume it's okay Friends, you are given a f u * v y and v you have two functions f given you u y and v are your two functions with respect to x means this is your function with some x this is a function of some x then f is an address what happens is simple what to do Keep one out and the other in find the derivative d body find one take it okay d body put out one plus one then take out the d body one off the other one or Say it or you can look at it like this Just keep listening in a simple way and understand everything I will come and look at it in a simple way, boss that y * v is the derivative of u * v the derivative of v plus the derivative of y You can remember it in this simple way Let me give you an example, I just did this I just did this, look where x sa x what did x do x went this has come now Do it directly, please pay attention Ok example example x sinx-cosx What is the derivative of x? Just explained look at x to the power of x to the power of n below do x to the power n - 1 what about x to the power 4 will become 4 raised to the power of 3 x what happens to the power of 10 bring 10 down to x to the power of 9 what about x to the power of 1 will be one brought down to the x power of 0 means this Your one has come, if you understand then x's derivative 1 so friends sinx-cosx + sinx-cosx x k x psa x that is the answer here p x k x p sa x understood okay so this Friends, you have to apply the product in this way that boss if in two function multiplication if they are written then keep one out and the other do the derivative plus put the other one out do the derivative of the first what is the division rule It is very simple, what does the division rule say? It's as if brother Both your and yours function with respect to x u up v you need to do the derivative of okay you need this d body one off you d body one off or body needs to be taken out ok How to find the direct formula, see what do it for the one below square the one below it Also below is the function, you square it Then you keep the one below outside, below this don't take this out but take this down outside d body one off do the top minus up keep the one out Ba d do a off down one think this way or you can do the same with the one below put the square bottom one outside of y do the derivative y d minus y put out v Derivatives of α okay let me ask you a question now There will be better clarity like this one is fine one of these is right You can see the example here as well, I told you take the derivative and you can see it a function above a function below a function above a The function below will apply a function division rule ok then eat fd directly comfortably square the bottom one Square it up. Keep the bottom one as it is. d Ba d one off do the top minus minus up one is written as it is d ba d one of these the one below is not this one below is ok let's just write x then the derivative of x + 1 what will be the derivative of x + 1 sibling derivatives d / what will happen to dx1 simple d / dx1 dx1 it's not okay to be scared of d / d if yes then what is the derivative of x and Ro is a derivative of forest so this is your forest So friends the one of x + 1 has arrived Your forest has arrived - x + 1 excluded derivative of x your It is one, right? That's right, even x + 1 has one value. 1 x is also yours and very good below your x So here it is your x - x - 1 / x so x went to x this is yours -1 / x s -1 / x you have your answer for derivative The derivative of it x + 1 / x is -1 / x ok let's do some more questions first You learn the chain rule, so what have you learned now? I took it boss that if we write two functions in addition If the derivative is applied outside then both of you Two functions can be derived separately We have to do their derivative written in subtext So you can get the separate derivatives of both the functions. You can subtract multiplication by doing I have written two functions so how to do it Write the derivative of two functions in division How to take it now what is chain rule Friends, the rule is used for the function Is there a function inside or is it composite You can also call a function, a function inside a function How to do it is very important function function inside function inside function now its I mean, let me give you an example the question is sa x s sa x s okay f gave you one it happens f is given to you x now this How to have a function inside a function Trigonometric function with x inside it Brother this x s this x s is of trigonometry only is the angle of sa is the angle of x is its You also understand the quadratic function inside Meaning of Quadratic in Trigonometry See if it was written like this x s * sinx-cosx your quadratic function inside of it How to do it comfortably: Start from the left start from left it's always ok ok what is the derivative of the x sine if Square of sine of angle Sine of angle sine of angle derivative cos of sine of angle of angle will be cos of angle san ka The derivative of x is the sum of x and its After that do the x s separately, then friends x s If your multiple is 2x then the final answer is 2x * K x S This is your derivative answer. got what i'm talking about okay now here look don't get confused if this it is written sa fxxx.pro Meaning in Trigonometry Trigonometry should start from the left f what will come what will happen to sa then Derivative of sine of angle What is the cross of angle sine of angle If you are able to understand the angle then what is the meaning of the angle off angle k off k x to sa k x k k x sa k x K K x Hey, you think about this x You think about this x x of x just put the x of x back understand what am I saying okay let's go Now what will be the multiplier of the remaining x of x The weight of the body is x sa ok so what is the answer - sa x in ka k x this is your derivative of The answer is okay, you still have more such questions. I will get better clarity, this is my I thought it has already happened, look at this sign The angle inside is also a function The angle inside the sine has a function And that angle also has a function, so does the angle there is also a log start from the left what would you do wud be the sign of the angle What is the cosine of angle and the sine of angle cos of angle and nothing else sine of angle what will happen to the cusp off angle then cusp off Understood the minus sine of the angle of the cosine of the angle okay and then lag x ka friends 1 up x so this is your final The answer is fine, you can write it properly yes brother this is not in minus is it in multiplication with this and with this All three things are used in multiplication This is how your chain works in multiplication The rule seems to be that a function inside a function is fine what will happen to it first go to the left side so there's something inside the log something inside the log is log of x is 1 over x okay it looks like x means 1 up x I understood it I understood it So whatever is inside the log, don't worry about it put 1 of yours sa k x s multiplied by inside the sign inside angle cy then what is the angle of will go to k off angle then k off c s Then what will happen to k x s? Will b of x s mine sa a s ok k x s ka mine sa s and then your 2x of x s so this is your The answer is everything in multiplication, everything Is it okay in multiplication, isn't it clear like this? Don't think brother that there is anything in this minus I don't have to solve it, I just look at you like this Okay, now this is an interesting one. bit cube x bit cube x I can call it like this to the power of x 3 means first of all you know friends x what happens to the power n of n raised to x n-1 this is what happens so 3 goes down 3 goes down Like x3 is multiplied by -1 now what will happen to x If k x goes then this will come out, basically what is the answer that 3 sine square x k 3 sine square x ok leave it if you want but this is ok easy is it power four what does it mean cos cos whole power phone of lag x this is what it means If it happens then it will be the Power Four first what will become power four power four came down lag cos lag cy one power 3 one power low Now the multiply by power has been tightened what is the minus sign of your cos Friends, whatever angle you write the minus sign It seems like something has happened, then what is it, what is it Apon to 1 apna sa ek fir sa ek ka friends K x then the answer will be so big and your answer will be like this Answers come here okay so chain rule also i hope you guys You must have understood, let us now follow all these chain rules Product rule is done, division rule is done Gone is the algebra of derivatives With the help of all these formulas, some very Now you have to ask basic questions like that brother what did you say simple take out the derivative So you know what is a function or a constant? Function into Function Plus Constant Function Nutu function constants make a big difference no you do x you know friends simple function plus function so If you take the derivatives of both separately, then f ' x what will happen if this is your fxxx.pro so what if the derivative of 2tanx will be 2 s x t x is s x to x to t x So from me x t x this is yours friends Answer of derivative Derivative means slope I am taking it out, basically I am taking out Snopes i am at x point ok how to do this How to do a very simple derivative of this tell me friends its function into function what did i tell you what did i tell you Suppose this is your y then what will become of y' Put one out 5x k + 3x - 1 or else Write it directly as d y / dx5 x + 3x - 1 out placed d / So boss, I am keeping this one outside 5x K + What is the derivative of 3x - 1 x - 1 of x Derivative 1 of -1 0 okay p x - 1 the same as 5x k I can take different derivatives of Lie algebra According to the derivative of it, it is three, right brother? Taking the hole derivative in function addition take different derivatives it makes sense awesome so 3 down 3 down friends 3 Power I told you power comes here look at power of x n n your power of x below reduce one so 3 down to the 3 5 15 15x power Reduce it by one plus this is 3x your friend If the power is 1 then the power of x means 1 a 1 - 0 So 1x to the power of 0 means this is your answer So 1 * 3 will become your 3, if you understand then this Your friends will come 3 Yours from the forest Ro easy now multiply it, if it comes to one only Multiply anything in one and the same thing comes out If we have to subtract 5x + 3x - 1 then x - 1 * this is x going into it so this is yours will be 15x k x going into this so this is your 3x -1 then - 15x is yours -3 Solve it easily 5x + 15x Will be 20x cubed 3x + 3x Will be 6x your minus ka 15x s -1 -3 e -4 correct yes this is correct this is the answer of your derivative is 5x a + 15x your result is 3x + 3x your result is 3x 1 - 3 - 4 everything is ok let's move on Let's go now, this too will happen, I think From Everything else is easy, friends, isn't it? Come on let's ask this one, a good question take it easy and see what you are saying What are you saying in the question? Public, come on, I will do one, you will do one Tell me which one I think I should do You may find this a little difficult. or this okay let's see let's do this are find the derivative okay so what the function beneath the function the function beneath the function below function function is ok friends d body of x to the power 5 minus a is equal to one what will happen to the one below turn the square down into a square up what to do you have to do first go down gonna write then d body one off body x off up the minus minus is to be written below One with the bottom means sinx-cosx and do d / dx6 and understand what is happening here okay the bottom one is out derivative of first one minus the first one kept the one above out kept the one above out derivative of second means the one below what happened friends sinx-cosx the power of x reduced to a power of x then plus x instead of minus because It's already okay here friends minus of x to the power of 5 minus of x to the power of one friends x becomes whole divided by sa square of one i basically sa square one So now multiply this by 5x The Power of 4 in sa x plus sa sc x - x to the power 5 of x - x power 5k x ps cos x plus co sc x whole divide ba san sc x now pay attention to the sa s x p k s x iv Here, your remaining 5x power is 4 bits. x minus - one to the power 4 of x a sinx-cosx It should be forest, I have not even written forest If yes then I made a mistake, write one here ok ok now see what you say we are finding the derivative of this bhai sahib At x = 1 now you have to find the derivative at any point But I asked him to take it out at some point To find the slope at any one point Told you to take out the derivative now general We are asking you to find out f ' at the point x f ' one is now asking you to find f-1 First understand the meaning of the question, I gave you f is 1+ x + x s + x k dot dot dot x k Power 50 told you f dove tell me told you f dove tell me the meaning derivative at point one tell me what do you do If you take out only f one then what will be the result of f one should I do it directly function ps function plus function p function dot dot dot plus function You can derive different algebras So by the derivative of one, 0 of x friends your 2x of x s 3 of x is below x s dot dot dot 50 down to the power of x 49 f'x is here hey boss so f-1 is easy no put v in place of x oh f-1 means x place and there is already 0 1 here 2 * 1 + 3 * 1's dot dot dot 50 * 1's bar 49 In that your 0 + 1 + 2 + 3 + 4 dot dot dot dot dr 50 will come okay so from one to 50 You have to plus till then, now how will you do this How will you do it brother, show me how to do it Can yes you can do arithmetic progression Ok Well, this will be pretty easy I think let me do it directly and see both of them The product is sinx-cosx The derivative of c x is - sinx-cosx If there is any other good question then I will take a look otherwise the question is almost same friends it's over okay let's do your homework I will do one more thing for you guys in your homework. And what is the division rule and answer in this tell me square the bottom one Let me do a little bit then x - t x square the bottom one down Write 'As It Is' in the corner two okay take out the derivative do it directly Let me take the derivative of the above one from x what happens to sec1 sec1 x and t x of to s x minus minus the one above like that like that Write down the derivative of sec1 x below so sec1 of x same sec1 x brother sec1 x and t x sec2 x I understood what I did once again I will tell the one below as it is written The derivative of the above is then sec1 x tan2 x Then the one above wrote 'as it is' to the one below derivative of sec1 x tan6 of sec2 x ok is there any cut or tear in this sec1 x sec1 is x is there any cutting or something But look at you, everyone has changed in one go If you want to do it then you will do it as homework I've worked out the derivative for you You will have to change this in your homework in two years and you please tell me that what is the answer It will come okay let's go with this if there is anything else If I have some time left, I will take a look, otherwise I think that roughly yours is here but this lecture ends it's very good okay it's been a long time There was a demand for this chapter, I hope that I am successful in telling you the meaning of limits Explain what derivatives mean what is it ok to explain it i am telling the truth I have been doing derivatives when I had to teach So I mean I have listened to lectures from many teachers. see why because the derivative comes Teachers know that it is very easy but Explaining the meaning is a different art but I have tried my best Try to give your best from your side friends I hope you understood it, please tell me how are you feeling if you have gone till now I have seen the lecture, right? I am on my side I have given you everything that you have It is important for school exams please don't practice any questions People should do it from NCERT, right? In NCERT There are a lot of good questions about limits, so I You will get all the questions from NCERT I have got some additional things done as well and for derivatives please you can refer NCERT The last exercise that you do Do the questions and do your calculations quickly it will get faster and nothing will be ok Currently your bus of class 11th calculus It has started, this is a very small part Your main calculus comes in 12th calculus if yes then we will meet there okay I will meet you again with take care of yourself until then and forever have fun, chill, just do what you want to do don't harm anyone tata ba bye love you all and take care by