Welcome to another mind map session on limit Continuity and Differentiable Limit Continuity Differentiable and keep in mind that this This chapter is very important as far as JE is concerned Maze and as well as Je advance is concerned If they are cubic then both are cube roots Both of them will then apply this and if a cube the root is a square root what will you do then Then a limit was placed on the whole question It will never happen that you have left half A limit has been set in the part and half part I did not put any limit in the greatest integer We put limits, they feel that way But the greatest inti does not keep limits we value the importance of food hello dear children As always, Jai Shri to everyone from my side Krishna Radhe Radhe Radhe Radhe to everyone and Welcome to another mind map session on limit Continuity and Differentiation means that today we limit continuity differentiable which these There is a chapter to decode it completely are gonna Thorates what you studied in your classes Here we tell you what you can do with it I will get the whole thing revised That is, what is going to happen here today is Limit of continuity A mind map, that is what we are going to do today Mind Map of Limit Continuity Mind Mapping each topic here in your mind is going to be mapped in limit one by one continue differentiable and keep this in mind The chapter is very important as far as JE is concerned Menge and Age Well as Je Advance Judge Concern In JE mains, questions on it would come up immediately. There are problems in JE advance as well are asked and mostly the problems which They are the greatest integer fractions There are a lot of problems with functions like parts These questions have been asked in JE Mains also Especially in advance, there are some different levels problems arise on which you can watch videos You can go to JE Challenger and see it there Advances of Limit Continuous Differentiable Some discussion of the problems is pending You definitely practiced in class If it happens then what are you waiting for, get one quickly Grab a notebook and a pen and get ready To make short notes, today we will tell you the best First, let's talk to the limit, what is the limit you Whose limit do you take out sir, limit do you take out whose limit is found by the function that where do we take it out at some point Do you usually take it out at which point? where the function is not defined or takes the indeterminate form and around it The neighborhood should be defined around or Let the function be defined in the neighbourhood then whose limit whose function's limit Brother, let's find out the limit of the function where is the function at a point where is the function at a Why do we take out the points because Function becomes function takes indeterminate Form at the Point Function Texts Indeterminate form indeterminate form at the point in determinate form at that point form at that point at that point takes the indeterminate form so we If we find the limit of a function, then the limit is one the way I told you It is an investigative tool with the help of which we Anyone can know the behavior of the function at the point where that ah what happens takes an indeterminate form or a defined form is not defined in its vicinity There should be seven indeterminate forms we have Did you read the Seven Indeterminate Forms? what was brother sen indeterminate Forms Indeterminate Forms Seven Indeterminate Forms Thi Se What were the indeterminate forms, Sir Zero? into infinity 0 by 0 infinity by Infinity not to the power infinity then infinity minus How much is infinity brother one two three cha pa more Tell me quickly which one was it yes the power of infinity is zero and 6 more left Now tell me which one else survived Sir Infinity Inu 0 to the power of infinity 0 0 to the power of 0 0 power 0 is saved sir power of 0 is zero so this is var T 3 4 5 6 Sen Indeterminate Forms and I told you every indeterminate form In you will find two opposite forces on the answer It works because of which the answer is settled it cannot be told what will happen these a forms indeterminate because we these Can't determine where the answer will go It will be settled, this is beyond our control You will have to do an evaluation, only then will you know It will look like this, it cannot be told by looking at it What are these called indeterminate forms? That is, it is defined but determined cannot be done and keep this in mind The meaning of zero is tending to zero, this is This is zero, this is not zero, it is simple zero What does tending to zero tending to two mean Zero means moving towards zero here What does tending to zero even mean what does it even mean tending to zero and This one is not the exact one, what is this tending to one is tending to one so these are The Seven Indeterminate Forms That We Have And whatever numbers you see here I am showing you the forest, my friend, I can see all this All the trending values are late not Exact Val Act One's Power Infinity The power of the ultimate one is infinite only one Is Exact Zero Into infinity exact zero equals zero into Infinity something tending to infinity will give you zero only remember this thing okay no and then I also told you that Upon Very small positive number naa aap very small Positive Numbers Tend to Cry Will Give You Plus Infinity and One Upon Very Small Negative Number Very Small Negative Number tending to 0 will give you minus Infinity, so these were some of the things that we told you about told about the lecture Indeterminate Forms During and Along with These are some important things, okay, consume it Indeterminate form is now off limit what does the function limit x mean tending to a function a function is said to have i said to have a limit of l a limit off L at x itu a if if a at x equals a at x equals a e equals to r h l at x equals a, that is, if a function limit we say l is at x equals a that occurs if If LHL & RH L R equal at x equals a and e equals to a e equal to a then the limit of any function we have a When do you say L on a point? When do you say A on a left? Both the hand limit and the right hand limit on a where l is a finite number is there a finite number i.e. total Overall, I can say if limit extending to a minus which We call LHL e equal to e equal to Limit extending to a ps f one that we The right hand limit says I'll be a finite It's quantity, suppose it's L, that fight What is the quantity less than limit x tending to a limit what do we say Extending to A of F one gets this l this means this means this SX Approach A SX Approach A From left and right From left and right and right the value of f is the value of f An approach l value of f approach oh okay that's what it means Its k f k x moves towards a whether Whether you move from left or right, just move One should take both the sides In which direction does the value of f increase with respect to a and if this value is towards l It does not increase, both give different answers So we say that limit does not exist or Then final it does not come, even then we say Limits do not exist, we do not say limit exists then in that case we will say That the limit does not exist in my opinion It must have become clear when we say Limit exists here, this thing is amazing It has been said here that this thing has been asked that the function is also to the left of that point is it defined or is it defined in right also It is assumed that both sides are defined That is, if the function a is equal to a point even if a but whether it is defined or not, to the right of A and If both the sides are defined on the left then this The definition applies but if the function is defined on only one side then it There is a slight change in the definition Let us discuss that for just two minutes. wait here But you must understand that this what is it sir this is LHL and what is this It's done sir it's done aa a l this wa l and This real y l and v real brother i am feeling it There's something wrong with the board here okay okay l and r now here's a important point important point is this Some important points that you should remember The first important thing is required Point Limit Extending to A Ace Employees x is not equal to a i.e. x is equal to a is increasing towards x is ever equal to a There is no second limit, this is clear x is not equal to a okay then the second we can talk of limit we can talk of limit at x equals A provided for wa if a Provided f a x is defined a x is defined in At least in at least one neighborhood at least one neighborhood nbd one Neighborhood means left or else right at least in one nbd of x = a v can talk of limit at x = a for y = fx8 one nbd of x = a such that l a of x Limit If someone asks whether there is such a limit at x = -1 If you take it out or something comes up then we'll talk about it can't go because in human's neighborhood There is no graph of A in Third For limit x tending to a a x u exist f may be defined may or may not May be defined and may not be defined at x equals a may and may not be defined at x = a means we can talk about limit there as well where the function is not even defined That means such points can also be discussed of limits which are not in his domain What is the condition in the neighborhood or in the neighborhood There should be a graph on either side i.e. the function should be defined then left should be in me or right or both defined in at least neighborhood so keep that in mind At least one is written that is why it is written so this There are three important points that will help you we will try to understand it better and then after that then after that I move forward, after this I move forward So this is about limit and limit What is Definition Limit Indeterminate What are the forms to remember in the limit what are the things okay you know all this well You must have understood, let's move forward now Next is next which we read Various Strategy Strategy to Evaluate Limit How to evaluate limits Let's see what the strategy is Strategy to Evaluate Limit Strategy to Evaluate Limit to Evaluate So how do we evaluate limits? So let us see, the first one was substitution By Factorization Factorization using algebraic identities the second one was release remember yes sir Is there rationalization or double rationalization? Yes double rationalization can also be done were able to do that as well were the third Substitutions Substitutions The fourth one was using Bano Miyal theorem and fifth we could expand These were the five ways to use expansion So that we can evaluate the limits and there were some basic laws that you can punch I told you about God's form Punch God God Five God I have given you five The limit for the properties was that If limit extending to a a becomes equal to l end limit Extending to a g becomes equal to one m where l and a are two finite numbers then Here we have made five important points I had read that this was the first limit extending to a f a plus or minus g a this equals is l p my m k i.e. limit a x tending to a f x plus minus limit Ex tending to ag x i.e. to the limit we can break the limits Distributors can but when The limits of both are finally different You do it and Fina Italy also exists Only then can you separate them, then we Had read limit ex tending to a single in gee one This equals limit ex tending to a f into a limit x tending to a g One and again it becomes l * m the product of the limits of both then the third which we read that we read was the third limit extending to a times f of a is a constant which is not a function of x, just say So this becomes k times a, that is k times limit is ex tending to a ex no yes it comes out i.e constant comes out constant ch in the product comes out of the limits sign then the fifth one that we I had read that if F 100 is is big if f a ig gun equal to g x then limit x tending to a a x fourth fourth here fourth a x i also greater than equal to limit ex tending to A.G. x x tending to AG x so this was fourth point so we have this four now fifth Let's also see what was the fifth one let us see that also what was the fifth come See the fifth point which was fifth The point was something like this Was ok let's fix it a bit here it is here and here it is here okay now the sixth point fifth point sorry What will be the Fifth Point? The Fifth Point was Points for the Division Was For what was the division The limit x tending to a f divided by a Byeji one this happens. My A and You are absolutely right, what is the limit x tending to a f one upon limit of g what should these two be in one division they go away they become different and the conditions what happens Provided Provided that the amount is not zero provided a is not equal to zero my Brother, it has become clear from my thought, yes sir this It became clear, I understood this it is clear Now the matter comes here, we move on to the next These are our five basic loss methods. brother write yes to this because in life Beauty has great importance limit Extending to an ex upon limit extending to A.G. X provided provided what is not equal To sow the five basic laws that we had use so these are the five basic laws if you Any violation of the five basic losses If you do this then you will get miraculous results there are full possibilities There is a possibility that you will get amazing results. If you observe these losses anywhere, you will go I did not do it so it is very difficult to observe these losses The five losses are more important then this then after that Later we told you the Bano Mial theorem I told you what is Bynum theorem 1 p x to power a equ p n a ps n in n mive upon 2 factorial x square in a myve upon 2 factorial 3 Factorial with n mite a minus u x cube so on till Infinity here, A is a natural number. there wasn't a whole number i should have said no and there was one more condition and one more condition what else it was there, remember it quickly, I remembered it, yes I remembered it Did you not come brother, I am remembering you a little sir what was the condition a little condition was mud x should be less 1 mud x e lesson 1 and sir you taught this in class also told that we applied this theorem even then just in case when x is the hole number What will happen to the terms automatically after a certain stage If everything goes on becoming zero then that means you saw If you go, this is what is written on the board, it is just should be applied when n is not the hole number and mud x lesson va ho we have written this but This can also be applied when n is the hole number because when n is the hole number, a stage is What will be the terms automatically after that If it starts becoming zero then what will happen next The expansion will not come, finish it in that number of terms will come then what do you mean if n is a natural number is a whole number then you nc0 nc1 a nc2 a s plus so on up to n s x of What is the power n if n is a natural number If the whole number becomes zero, then it will reach zero Only value one will come, only the first term will come ok but this is the expansion applied here also because nc1 is n this can also be done We told you AC means n in how many factors r from n-1 to n-2 Factors divided by r factorial is ok the only difference is in this What is the fight number of terms in this infa number of terms come in it but If you want to know if n belongs to a natural number then this even if you put this condition it will still work Because after a while you automatically move forward The terms will become zero, we discussed this in class I explained it to you in detail as well ok so it's done brother Then in the cubic I gave you a In factorization it was said that anytime If it is written cube root a plus cube root b then If you multiply it to the power of a 2/3 plus b to the power of 2/3 plus not minus a to the power of 1/3 in b to the power of 1/3 so this is a If P B happens then both will be included in the rationalization cubes are cubic roots then we can You can use the minus cube root of If b is there then you multiply it by which a a to the power of 2/3 plus b to the power of 2/3 plus a power 1/3 of b to the power 1/3 and this will become a - b We use this in evaluating limits we do it in problems okay right and if both are power square root then you get We know that rationing will become double ration ization will happen if both are cubic Both are cube roots, then we can apply this and if is a cube root then is a square root Then what will we do, then we use the expansion will do the Banno Mial theorem okay and The third important thing I told you it was that limit x tending to a limit ke karan arjun - power of a n up x - a is n * a to the power n - 1, and In the same way, the limit x tending to a x a to the power of a - a to the power of a to the power of x m - m to the power of a becomes n * a power n - 1 to the power m * a -1 and this becomes m up to the power of n a n-- You may not remember the one with m seconds, but You should remember the first one, then another And the important thing which we have told you in limits It was said that it was limit n tending to limit x tending to infinity and x p a1 e x p a2 am I remembering something yes sir I am remembering Sir, I am remembering this one, yes son something I remember, yes I remember sir x P A1 one p a2 to x p a to the power 1 by a this is the whole power 1 by a yes brother this is the whole power of and by a minus x My one and we have to find its limit What should we say about this brother Find its limit, yes, quickly tell me what was it sorry what was its limit Arithmetic meaning of a1 a2 a that was it this we told you I told you we had already done this in class And this was an important point, right? Yes, I remember, and then I read it Trigonometric Limits Trigonometric Limits Let's read what we saw in limits Let's see what we did in trigonometry We had read about the limits Tragus limits tragus Limits In trigonometric limits we have Let's see what you read Trigonometric Limits We Did the Following things what did we do what did we read brother Trigonometric Limits The first thing we read was what did you read sa x my x's limit Limit Extending to zero sa x upon x sir this is one and this We wrote the limit x tending to zero x your sign x and this was written limit x tending Two zero one cosec x x cosec x is equal to limit x tending to zero sine times the weight of x upon x is equal to limit x tending tattoo zero x your sign Ewers x all of this was one all of this was the limit how much was this one then Secondly, in the same way we read for no also. it was for that what did I read brother it was for that we had read it for vis not also we didn't read it I had studied for this also, what had I studied for So I had read limit extending to for no Zero one and am I remembering anything else yes sir I am remembering something I must be remembering something brother yes I remember what I remember brother limit x tending to zero x upon n x and limit extending to zero coat x x cut x and limit extending to Zero Limit X tending to zero never of one's own upon one and e equal to limit extending g x upon never of all these limits which were all these There was a forest and one thing to note here Golden Points: Some people told me about Golden Points it was the first golden point the first golden point was sign x your x in Vicinity vicinity of x equal to 0 Sign Up x is always between zero and one Where it is always between zero and one I have what we have V Have you ever seen your ex? Is this a lesson or a lesson from zero to one? in between but the number will be something like this It will be 0 p 9 9 9, the number will be something like this 9 8 There is nothing like this in the forest It will be very close when you are very close to zero If so, it would be very close to number one. but it will be smaller than one then the other Baat Sa x In Vicinity of x e 0 V have what do we have t x Upon X it will be bigger than one it will always be This will be bigger than one forest it will be bigger than the forest it will be bigger than the forest means it is a little bigger than a forest It will just be 1.00 one then any number like that There will be a small number near one Good Sign in Ever After Your Ex would like to say something about sign evers of x upon x here sign evers x upon Ekas it will be bigger than the forest it will be bigger than the forest and here the evers of x upon x It will be smaller than one Meaning of smaller than one same here 999 something exactly one If you are in the neighborhood then you will know both these things I have to remember the third one The thing was that no matter how many of the above two The formula is now in now Results in Abba what should be the results x should be in radian x should be in radian this radian to be in x should be in radians and Third, there are two trigonometric limits, We have read a third limit which we I had read that limit is x di to zero 1 minus cos one upon x square i equal to Half this was also an important limit 1 mic x up x s what is its limit half so this was the important trigonometric limits that we had done isn't it There were important trigonometric limits These are important trigonometric limits And whatever we used in it, we I told you here, now let's talk At Let's Talk Expo at Shial Expo Shiel End Logarithmic Expo Dansh And Lagam Limits Logarithmic Limits Exponential and Logarithmic limits let's talk about it let us talk off Expo Logarithmic Limits Let's Talk About It them let's talk about sus expo mentioned Logarithmic Limits Let's take a look at Expo Logarithmic Limits What we read in Expo and In logarithmic limits we studied the most First Limit Extending to Zero e to the power x minus v upon x this wasn't it and limit extending to zero l a ofv ps x upon x this also n was and the limit extending to zero 1 ps x to the power n its x was Or we can also say it like this, limit x tending to infinity 1 plus n my ex ki You can also say that it was Power X or who C form was the power of one was infinity form see yes sir this one to the power infinity form if you were there then definitely The Three Important Limits TV did we read another one which one was quickly tell me which one did you miss yes he is right You are right my friend, I read this form the limit extending to Zero Extending To zero a to the power x minus v upon x this thi l n of a this was a a of a then we I read another one here that we call Generalized One to the Power Infiniti Generalize One Ha Generalize One to the Power Infinity remember we read about generalization to the The power of infinity is generalized to one power infinity form in this it was if limit X tending to a ece this is one ho and Limit Extending to a g x is infinity plus or minus either of the two then extending to afx to the power gi x what is this this becomes to the Power limit extending to Ax comes forward and F is one of the one of the subs gets subcut sot it that was not generalized to the power infinity form it was generalize one to the power infinity form to remember do you remember right Then I will look at one important thing All the important things that we discussed in class I am telling you here that too Along with this, another important thing is that I told you in class that Whenever the limit becomes non-zero and you get any In that case, you cannot see the way limit extending to a if anything is written you put in x = a + t Substitute x as a + t and then proceed go ahead and then move on okay one more thing a here But somewhere here the kids will say Sir Left hand limit right hand limit you told me that both should be equal what is the function There should be function on both sides If it is defined then you have told LHL and RHL should be equal but you have did not say that if there is only one function What to do if you are defined on point So here we will tell you that as well. let me tell you So one sided limit one sided Limit One Sided Limit One Sided Limit, just look at what is one sided limit Brother in one sided limit You need to look at this function only if function is defined only in if a function is defined only in n bd of x eq AV use one sided limit i use one sided Limit means that if there is only one function Be defined on the side, in the same neighborhood be defined here But did we tell you here if The function is defined only on the left side f e defined only in left neighbourhood i wa i a eks is defined only in left n bd of x e a limit x tending to AFX will be equal to limit extending to a minus of f x that is The left hand limit will become his limit If the function is defined in just one place On the left side i.e. limit a tending to zero a of a minus a where h over zero it's big okay and in the same way If wa e f one is defined is defined only in right NBD Right Neighborhood of X E A then what was the limit ex tending to a f one this is going to be equal to the limit extending to a ps f x i equal to limit x tending to h tending to zero f of a p h and h will always take care is greater than zero whenever you take the LHL or If you find out anything from RHL These two things were for one sided limit sided limit I think you will find it very useful You must have understood well that one sided When to use the limit, we are talking about two sided When do we talk about two sided limit okay so when the function is on both sides it is defined then we use left as well edge right hand limit and when on one side If defined, only the left hand limit Or you could have managed with the right hand limit are ok now this is expo limits and I also told you to remember all of this the way to keep it was to the power of e something minus and upon something and where is that something should go to zero so the value should go to one sign of something upon something will come and that If something is going to zero then we say can limit one I will remember this thing, then the next one was Using Expansion Limits The next thing we need to read is expansion Limit Using Expansion Limit Using Expansion Expansion Here we have given you lots of We told you about a lot of expansions tell me the expansion the limit using expansion so we had done lots many expansions in class but still You can go through all those expansions at once. take a look at the notes right now right now and like I told you this I have given you the limit of trigonometry if you want. Some of these SR cues are still with Try it with this and the chapter will be revised automatically The entire SRQ that was in it will go ok i told you in the expansion Some important expansions on which I have I also marked four was the first is sinx-cosx factorial x to the power 5 upon 5 factorial x to the power 7 upon 7 Factorial Son Till Infinity is the expansion of a single in this way even powers comes na mine x sc up 2 factorial x to the power 4 up 4 factorial x to the power 6 upon 6 Factorial Until Expansion to infinity and then none The expansion of x was one plus x well up 3 ps 2 up 15 x to the power 5 so on till infinity then e to the power of x and a of ex The expansion was also and p x 1 p x p x spen 2 photols ps x spen 3 photols Til Infinity and A1X expansion this tha a x minus x plus 2 p x q up 3 minus x to the power 4 up 4 on till infinity till infinity so this was the expansion of aa ofv p a ye eva p a ka expansion that was expansion of l 1 p x These were some important expansions that you definitely remember I want these four or five expansions which you should remember you should know them these days Some of the important expansions, however, In class I told you about Never Say Never Ivers of Ex, we have done all these expansions I have told you about all these expansions I've told you so if you want you can Go to the bar and write all those notes in your class You can see the expansion here, it would be nice ok now there is one more important thing for you What I want you to remember is what is the important expansion brother it is 1 The Power of PsX yes and you also saw that it is used The question of flour is asked in many questions did you remember it was used in this He was not plus The power of X is not a p This was one, yes, remember it and tell me quickly tha e in 1 my aces tell me if e is nine minus x is two I don't even remember brother P 11 Ba 24 X s 11 ba 24 x sk but I told you Even without this, I had asked a question and told you 21/4 x Cube Son Til Infinity 48 48 below is 48 brother yes I mean this lie I won't say it, I don't even remember Brother 41 by 21 by 48 but even without this It works, it works without it yes i even showed you how to drive you could have Done questions without using this expansion ok so I have told you about this already okay then we have some here Golden Limits I had read some Golden I had read some limits with the help of these Golden Limits and What Is Golden missing the limits some yes so the first golden Limit Golden limits so the first golden limit was It was something like this the limit is ex tending to zero n x upon x cube n x minus x upon x cube this was it na Byth then the limit x tending to zero x minus sign one upon x k this thi 1 ba The one with power x of s is not that important If it is there then you can take it out by expanding it yes you can easily apply and remove this also But the point to remember here is that x is x 3x is needed for the peace of the soul of is of the thing x and t is of the soul of x - x Peace also requires 3x and Then I also told you that limit x tending to 0 x psa x up x it becomes two na x p sa x soul of only one x is needed for peace Limit extending to zero t x p x of Just one x for peace of soul and this gives you to again then you should also remember this you should also remember this Then let's talk about your favorite, your friend Who is Manjulika about the function? If we talk about function here Do this, where do you use this expansion It is mostly used when you see below The power is more visible and you can see it above If less power is required then such better under the circumstances If you expand it you will see that If some terms in it get crossed then As much power is available below, same power is available above is also available and the limit exists If you go then tell me about Manjulika And the fish is the queen of water, yes sir I remembered Sir, you meant I told you everything one by one, Sir, yes son I will tell you each and every thing f I am equal to you X Osit how does this function osit happens Osit Between-1 and One in Vicinity Of In Vicinity off one equals to 0 this is what we told you And there's only one way to control it He is Akshay Kumar and Akshay of Mathematics kumar kaun sir zero this will be zero isn't it the limit extending to infinity then You should see this Manjulika sign first This Manjulika is a very decent function It is formed only when the angle inside it is Where should he go from zero to infinity approaches infinity, then if Suppose it is written like this, saw up x then brother In this case, the limit will be extended to infinity saw up x up 1 up one more now look look this is going to zero and this is going to is zero then this limit will reach one, then this remember there are a lot of kids in this He often makes mistakes and doesn't see it we find that this limit will be one okay Then again I told you that the fish is the queen of water this is the concept yes sir i am remembering something I am remembering something son, it sounds familiar yes that means If any term is given above or below the given term, then one needs a term to have an ex If necessary, what should be done in that Use only that which is minimum requirement for example limit x tending to zero is written somewhere or cube One plus x my sai one more plus x squa plus sign squa x such written down and jotted down be x minus x plus and minus cos x plus sine cube X You have to take its limit to zero So you can understand its requirement 3 above one of its requirement is 3 one of its Requires two x's Requirement is also for two x's its 3 x its two x its three aces so minimum What is the minimum requirement requirement is up and down which is equal to t so up down we're gonna do it with x square we will divide it, with whom will we divide it x square se va will divide by x square we divide it by x square If we give it up and down then it is divided like this here Divided here and divided here this limit will become zero i think you You must have understood the limit extending to No Zero x n sk x your x sign x ps x my sa x upon x why in x then plus 1 ps cy x upon the whole square of x and what will be the bottom will be down not x minus x upon x cubed in x psv minus cos x upon x [ __ ] and plus sign x into inn y inn x will come inn x and here the sign x upon x has a hole square in sign x and you can see this will go to zero this will go to zero because these x will be up 2 and it will be half below so the answer is in this The case will be four so in this example it is I told you that when it happens that If the requirements above and below are different We should pay attention to the minimum is there any requirement yes then I have saved one law for you still what was that Sir Varys In strategy I was thinking you mentioned the Law of Why didn't you tell me about love? Yes son, we just said that the law of was kept saved for what is the law of love Tha Law of Love Law of Love Tha We Love to Create we love to Create Infinity Inn Denominator Love To Create Infinity Inn denominator in Denominator and and zero in numeration and zero in Numeration What does it mean to create something in zero infinity in the denominator and zero always Trying to create in numeration because under whose chair It seems like infinity, even god cannot save him could it tend to cry and whose skull If 'weeping' is written on his forehead then he will surely weep yes i was thinking this from the beginning sir I have not told you yet, yes son this was it and In this I told you that limit x tending to infinity fxxx.pro Paulino Inn x paulino mial in x so in this we have I read the first case that if Degree Degree of g x e greater than degree f 10°F is equal to one, so if I call its limit L two will be zero A will become zero and degree F if greater than degree G X In that case the limit that will come out is out of infinity or minus infinity Something will come depending on the case, you will see It will be fine yes if the degree is higher the degree is down and what if if the degrees are equal then the third is if degrees become equal if degrees a equals one degree g x to the degree of f is equal to the degree of g x to the degree of If it happens then I will b Let A be the coefficient of x power a in a unit Divided by Coefficient of x to the power a in g x in g g x t was the law of love so this was the law of love right we love to create infinity in noun r and Zero in Numeration And yes, now let me tell you one more important thing here. What I tell you is that you should never cross the limit Should not be applied partially, never apply Limit Partially Never Apply Limit Partially Never Apply Limit partially limit partially means whenever the question If I keep a limit then the entire All in one go in the question Limits will be placed on the question, will this ever happen It will not happen that you have limited it in half part kept it and did not keep limit in Radhe part If you do this, you will get amazing results there is every possibility of getting This is an important point which I want to tell you. I had to tell you about the second function with built in limit Let me tell you that if the function With built in limit given function Give a limit in advance to what is given If it is kept then in that case function with built in No limit Function with built in Limit Function with function built in Limit built in limit means if you Give limit beforehand in the function in question laid there is already given limit inside Assume the function is something like this evaluate evaluate Limit Extending to Let us assume that this limit is to be evaluated to be told and with In if f x is equal to limit a tending to infinity of something something so what should you do this First define the function in the problem in nbd of x ectu a i.e you this limit will evaluate f will evaluate a When you evaluate this limit here Then we will write when a to a plus delta k is in the middle and then write this limit when x is between a minus delta to a and then you find the limit on a so that is How You Deal With Problems That Are Built In Limit means if there is already a limit is inside it and if anything is written here You can first define it in the neighborhood of a we will do it because when the limit is talked about If so, then the neighborhood is discussed only then We are able to talk about the limit, that is Define the function to the right of the point do the function on the left side of that point Then there is a confusion that often arises Children love that it is the greatest int We are in the greatest integer regarding the jar They feel that they keep a limit But the greatest inti does not keep limits we value something like suppose sometime it is written limit h tending to 0 bit h plus sa yes let's go sa h plus greatest Integer of t to the power 4h and upon h then How can I validate my limit quickly? So you tell me what will be its limit h tending to zero this will happen a upon h this will be one actually many People think that here you We have kept partial limit, we are partial We are not keeping any limit on the greatest engineer If you are keeping value then be careful here Look at t to the power 4h when h goes towards zero. If this number goes to the neighborhood of zero This number must be a little bigger than zero, right? The number will be slightly bigger than the number row And when this number is a little bit bigger than the row So the value of its greatest integer will come and what will come, if it will come 0 then hence sin50 you have been kept h do not think at all that Partial limit has been imposed in the question Partial limit has not been imposed here The value of the greatest integer is used The value of the greatest integer is used I hope you understand this subtle point ok bro this should be clear So this little in this point also here this small question There is a lot to learn in meditation too Keep it and if fraction x ever comes, it will be more It would be better if you call fraction x the greatest change it to integer x minus greatest using the formula for integer x so that is limit now let us come over to continuity now Let's talk about continuity in Let Us Talk Continuity Now if we talk about continuity First of all a function in continuity is called a function When is it called continuous at a point Continuity, should anyone talk about continuity? When is a function continuous at a point then the function f is said to be Continuous A The function y i f a is said to be Continuous T to Be Continuous at x equals A If lhl e equal to r hl i equal to A so that means limit h tending Two zero f of a minus a is equal to the limit a tending to zero a tending to zero a of a ps a this becomes equal to F of A i.e. K lhl equals rhl ho hle equal to rh l e equal to f t is lh l ka means a of a minus e equal to a of a plus equals a of a e e a and if you want then like this you can also write How to limit extending to a mine of f e equal to limit extending to a + off fx8 continuity so we can check any The function is continuous at some point or not true if L AA Ra RHL = A or Then the left hand limit at x tending to a is equal to right hand limit i it a ho That is, it is the same thing, all of it is the same thing it is written in different way so if this If all the three conditions are satisfied then we Let's say the function is continuous on A then this Keep this thing in mind when it is said that someone The function is continuous at some point here is an important thing, this is an important thing we are the continuity first important thing first The important point is if A is continuous At hey look this is what sets it apart from the limit If f is continuous at x = a then a is defined at x = a defined at x = a but not the Converse Not The Conversely, if A is continuous on A So the function will definitely be defined on A. But if is defined on A then its It is not necessary to be continuous secondly Importent The second important thing is that we can talk Off Continuity Continuity at x equals a only if x is a belongs to the domain of a i.e. its We can only talk in this domain, this is the only There is a very subtle difference between Between continuity and limit we find limit of We can also talk about those points where he There is no hope of point function being in the domain should be a graph in the neighborhood but We will talk about continuity only when someone Only then can we talk about continuity at point A. when that point is in the domain of the function If I am there, if not, then it's a matter of continuity Can't do any more discontinuity there a matter of can't talk of dis Continuity Dis continuity at x equals two A Only If F Sss A Graph In Vicinity of x = a is there, then what can we go and say Whether a function is discontinuous or not Like how is discontinuous on na pa baat even if Pa Batu is not in the domain he is not in the pi batu domain but there discontinuity can be discussed why will we say brother a but not on the left Left hand limit and right hand limit both are not equal one is infinity one if it is minus infinity then it is not finite Limit does not exist then function If it is not continuous then we can talk off discontinuity t x e a only if a has a Graph in viscosity of x e a x e a around What should happen to the function in the neighborhood There should be a graph only then we can see the discontinuity can talk about Well I can also say this to all of these Combining all the points same thing that is limit x tending to a f must be equal to a should be equal to A Here we should put one more thing before you can they do this if a is continuous at x Equal a continuous at x equals a this implies limit ex tending to a f one exist final Italy is a fight but not the Converse not the converse i.e. if the limit If you exist then it is not necessary at all that the function is continuous but if If the function is continuous then it is necessary that Limit Existence will definitely fix it if yes then this is about continuity talk again Continuity So how do I check the continuity? They will do it, we will see how it happens It often happens that children get a little more The problem that comes is that sir limit When do we limit our left hand and our right hand Limit and when to withdraw direct limit we will give it in continuity because its his only If the application continues further, then any The left hand right hand limit of the function is then If you have any doubts, you should check it out that in the neighbourhood of that point the function will behave differently and some such functions those who behave differently in some In the neighbourhood of the point it is like Greatest Initial ... Behave differently in Inti's neighborhood Fraction does this also happen in some way or the other in the neighborhood of an engineer differently behaves like the sign of x this function is zero behaves differently in the neighbourhood okay so there is some function like this inverse of sa x is sa inverse of sa x cos ivers of co a n ivers of n x this is some There is a function that behaves slightly differently if we do it in the neighbourhood of some point then We would be better off with such functions that We take out LHL, RHL and then go to the limit Please comment in case of such function Correct if so Now there are several types of discontinuity discontinuity here first one more thing tell me the name Function Paulino Mial function expo Function Expo ctional function logarithmic function logarithmic Function and all Trigonometric functions R Continuous in There domains are continuous in their domain and I also gave you an example in class. don't assume that after reading this that every function in the world is in its own domain Don't have such misconception that it happens continuously Take the example I gave you, who Write it in the comment box below Yes, I quickly gave an example too Class I so Now let's talk about discontinuity yes first continuity Let's talk about these intervals, continuity? in an interval so now we will be talking of Continuity in an interval will be talking Off Continuity continuity in an interval If we talk about continuity and interval When is a function continuous over an interval? Selecting colors will be called a big deal it is difficult bro function wa e a x is Continuous in Open A to B if it is Continuous if it is Continuous at each and every point at each and avery point of open a to B of open A to B will remember this thing Secondly, when will a function be in a close interval? It will be called continuous, it is worth seeing a Function a is Continuous in closed A to B If the first The first and the most important point to remember F is continuous in open a to B in open A two second point f is this called right Continuous Right Continuous at x equals a that is limit x tending to a plus of f one should be equal to A i.e. right Hand limit should be equal to A K and A K should be equal to okay so function Right on A will be called Continuous, then right must be continuous and be a is left Continuous Eight B Left Continuous at x = p that is limit x tending to b the minus of f a should be equal to f of p this then the function v must be continuous on will become close interval a to b It will become continuous, this is the condition When we talk about close A to B then B can't you see what's happening next AK don't see what is happening first, only look From A to B means A and after A, B and from B first one is clear Now let's talk about discontinuity now let's talk about this Continuity Dis Continuity Dis continuity talk of dis Continuity of three types and two types Discontent yes which brother removable disk Continuity Removable Dis Continuity Removable Dis Continuity What is Removal Discontinuity it is here But here limit x tending to afx Exists Fina Italy Exists Fina Italy Butt or a is not defined A is not defined either A itself is not defined will be and it is not equal to A Not Equal To A is not equal to A i.e. either A is defined it won't happen or what will happen here is the limit Either it will exist or it will not be defined Or A will be defined but it will be not equal to limit and i will not be equal to a a on this basis But it is divided into two categories And what are those two categories We can see this in two particular forms let's classify it further and the two What are the forms? Let's take a look at one. So is an isolated point discontinuity isolated Point Dis Cont What is Isolated Point Discontinuity here l h l at x equals two A L H L E equal to R H L E equal to a fine it quantity but is not equal to A of A means the point where the function It is discontinuous, what happens at that point yes a real is equal but it is equal to a are not equal and appear isolated it appears that bhaiya a but the one who If the value is isolated from the graph then What do we call this kind of discontinuity? It is called Removable Isolated Point Removable Discontinuity and then missing point Discontinuity Missing Point Discontinuity what is this missing point disease In Continuity here LHL is equal to R al is equal to a fine it quantity but A is not defined but in this case a is not defined This discontinuity that happens is called The Missing Point Because what can we do about these discontinuities? I can remove it Discontinuities can be removed can be removed by redefining by redefining The function at x = a , that is, you can calculate the function redefine where x is on a Redefine the function, i.e., A. Take the value which is equal to the limit value then this function will become continuous So we call this kind of discontinuity What is Removable Discontinuity? So focus on removable discontinuity Keep the limit, LHL also exists RHL will assist each other Fina will be equal in Italy and they will be equal to each other but in this case either A will not be defined if A is not defined then missing point or else The limit that comes will be equal to limit A. If it is not so, then in that case The isolated point discontinuity will be u will have an isolated point discontinuity in this case i hope this It has been cleared It is very good, I understood the removal Discontinuity Isolated Point The missing point comes again non removable discontinuity next comes in non Removable Discontinuity Non Removable Discontinuity Non Removable Dis Continuity Non Removable Discontinuity is a discontinuity that we call Can't remove this function from the DNA this is such that this dis of function Continuity cannot be removed, i.e. here But the limit of the function itself does not exist So here limit x tending to a f one does not Exist does not exist and based on this We classify it into three different Forms in three different forms R two forms let's see one by one that's all what are all the forms yes so the first form comes in this which one is yours Type Fina it fine it type non removable discontinuity final it type non Removable dis continuity now lets see what do we mean bye fine it kinda happens here What is here both LHL And R H L R Fine it but Unequal Butt Both are unequal but are unequal like if I say that Extending the Limit to a minus a x this is done L and the limit extending to a plus of a one time it happened a this is done a so this function l is not equal to AL and A is not equal then y f Type discontinuity is fight type non Removal discontinuity and here's what we're talking about will say Whatever jump happens will be a combination of both There will be a difference, for example, suppose this is five. If there is three then A5 minus 3 will jump to two That means the function has jumped in a way, right? The function has jumped from a l and a to a and A and L to A depending on who is bigger who is younger then this is final it type and this is called its jump the second one is Infineon is infinity and minus infinity is not finite write it like this infa means either l h l ps infinity There may be an RL fight, that is At least one of the two limits is infinity should be one plus infinity should become one If it becomes minus infinity, this will also work infinity and this one will also run infinity it will minus infinity or minus infinity it will work minus infinity this finna it will work then this Whatever kind of discontinuities there are is called infinitesimal limit x tending to a fx does not Exist s function is Sussite E Citri in Vicinity of a like sign and your x at x tending to zero this has an alphabetical type discontinuity at x e 0 then where the function Which thing do you see ositing there The discontinuity will be of the type The discontinuity will be subset type Discontinuity Then there will be something important things even Difference Sum Difference End Product of Two Co. f off Two Continuous The function is Add two continuous functions If you subtract or multiply, the resulting The function will come and that too is continuous will be Then If F xg x R Both Continuous at x = a then and g not equal to 0 Then f one our g one is also Continuous is also Continuous at x equal to x e a now this too will continue x e On a Then if f x is Continuous at ex ect A&G X is discontinuous Continuous at ex equiv A than very important point is than F x plus minus gi x is also discontinuous i also dis Continuous at x equal to a & x e a Should p minus x not become a point function? do not become provided a x plus minus g g x does not become a point function does not becomes a point I gave an example in function access class f one ho rat one g one ho - r one rat off - one And when we add both of them, we get rate x rate - g This is a function which will be created by combining both Its domain will be only zero and zero's If it is not defined anywhere else, then it the function which is the point function then focus Keep in mind that any number of point functions all those functions Continuous is taken because in that case The definition of continuity in Rat Hand limit is equal to left hand limit that is If the right hand and the left hand don't match then Only one thing is found that A defined then in that case a should be defined then the function becomes continuous on a then all point functions are continuous if a function is continuous is a point function at an end it is then it will be continuous At a means that it is defined to the right of a Isn't it so that to the left of a it is defined only as a is defined on then the function is defined on a is considered continuous because that To create the function you just need to use a pencil. You will have to put a dot somewhere to end the graph It wasn't broken nor did I put a bindi but where is the bindi It will break, so it can be thought that If you are a continuous If function and is a discontinuous function then If you add or subtract both, it is always one You will get only discontinuous function then next If f is continuous and g is discontinuous if f is Continuous F is Continuous at x equals two and at x e a and g is dis continuous at x e A Dis Continuous at x ictu a then nothing Definite then nothing definitely can be said definitely can be said can be said about there About the Product their product and you will have to do a scent test and The question then becomes about their product and question Nothing can be said about testing you will have to test continuity now and differentiable about the continuity I am not talking about defense and in this We told you an unknown result, if X is a one in gi ek how is this function this function Continuous is a But Continuous at x e a but and there is a zero along with it there is a zero yes and it Function Discontinuous Ho Dis Continuous at x equal to a a x e A with Fina it lh l and RHL&G of A is defined and g of a what should be g of a should also be defined and g of a is also Now in a situation like we have defined atoning result where did fx become Continuous is continuous at x equals a then f x continuous will be equal to x But this was our atoning result This Vaz Atoning Result That V had this is the atoning If the result of a function is discontinuous and its lhl rhl is fine it and g of a is also defined and the other function what is special about it It is continuous on A, but it is continuous on A. No matter how much is its value coming on Sir A If its value is also coming out to be zero then we say you can say that the product function will be will also become a continuous function so weight atoning result tv did then its after then after this if both F and G be discontinuous if f A&G x are both dis Continuous are both dis Continuous at x equal to a then nothing definitely can be said then nothing definitely can be said Can Be said about late Product About There Even Difference Difference Product and Quiet Product And anything from ct and ect au canon V can go to Sarnia if both of them is discontinuous x equals a But then we can't say anything after checking us You will have to tell everything then we have put a thing and read it We read about tha composite function was about composite function Compositae in Read about Composite Function so what did you read in composite function Tha Composite In the function we saw if a is Continuous if a is continuous at x equals is and ji is dis continuous and ji is Continuous g is Continuous at x equals a a donation goof is Continuous at x ect a t gof Continuous it becomes at x e a i.e. if g Continuous is a Continuous is a on inside one on A and G outside on A If it is Continuous then the gof Continuous will be at x = a if a and g are both Continuous are both Continuous at x equals A donation Gough I and I shall not be I and I shall not be Continuous may and may not be Continuous at x can be equally continuous and If it is not possible then be careful inside is a continuous function at A and is outside The function f of A must be continuous on And then we told you in this that it is probable yes assume f is discontinuous if f is discontinuous discontinuous Discontinuous At X Equals Emma Insurance C&G is discontinuous Dis Continuous at x equals alpha beta gamma and so on then probable points of dis Continuity Given Probable Points of Dis Continuity Points of Dis Continuity R Dos r x e alpha sorry B C and points wear no problem pot disc point wear fx became alpha beta gamma and so on to Probable point means this is the probable Points of Discontinuity of Gough isn't it, should we write off goof like this x of gof of aces r those r wer f one this Probable points means these points It will be discontinuous if it happens then it is there and If these points That means if someone tells you Goff's points Of discontinuity tell me then goff's point off discontinuity what tests do you do we will check those points where whether the inside is discontinuous or inside The function taking on values such that the outside becomes discontinuous, then we We will just look at that point and go away if that The function is continuous at the points where If it happens then the guff continues everywhere you will remember this thing you will remember this thing We will keep using this, this is Signum's Signum Where was discontinuous Signum A Eks Sigma A Eks This is the sinum if we assume A is continuous e a e Continuous If F is Continuous Yes Sir You were waiting for this for so long F Continuous is then sign a where will be discontinuous Tell me then signum f a is discontinuous discontinuous Ate look now there is a goof so where is this will be discontinuous where the inner one discontinuous or internal The function takes such values where the outer one If the function is discontinuous then There is no discontinuity point with fx2 If not, then which point should we look at? Where fx9 where is discontinuous row at at fxxx.pro f a e 0 Provided Provided Va i f a does not know inside does not know Inside With X Axis with X axis in vicinity off let's say from here we get one equals AB: This is how you get points in vicinity of x equals a b and so on so this be About Gaya Signum i.e. Signum of f There will be a discontinuous place where the inner function is zero go away it will be discontinuous there The condition is that where it is becoming zero The whole of it is in his neighborhood does not lie on the x axis because if it If it lies on the axis then the value at A is zero means the value zero at a is on the right hand side of a Limit is also zero Left hand limit is also zero So how will the function be continuous will go i.e. it should not become zero constantly The Vicinity of the Points Where It Became zero ok yes sir greatest integer of fxxx.pro then the greatest integer of f one is where discontinuous is discontinuous is this continuous is dis Continuous at Points where x became an entity If you become an integer then what will become of it becomes an integer becomes an integer Provided Provided provided we do not get i do not have a Minimum Attendance point at point there should not be minimum because if it becomes minimum there then if you By making a graph of his greatest integer If you look at it, you will see a straight line there. if it goes then the function will break there otherwise The function will not break there wow sir one thing Everything has been revised in a great way The concept of limit is very close to continuity It has been revised very well till here If you have revised it well then you will reach here So it is clear, yes sir, everything is in continuity something became clear greatest integer sig And all this in continuity in some interval I understood it well ok No, this happened, then we'll start again So now let's talk about the differentiable over differentiable so we have continuity so Having read about continuity and limit, we have I have revised the whole thing in a systematic way Now let's talk about differentiables of differentiable Differentiable Now We Talk of Differentiable look it is well written It has no difference d i double f E R Differentia poly difference ability yes now it is written correctly now we are talking about differentiable then What does differentiable mean Differentiable means that the function what happens at any point is differentiable i.e. at that point there is a fin it slope of What can we make? If we can make tenge then The function is said to be differentiable at a point to let us see what do we mean to you function a function y = is said to be Derivable Said to Be Derivable Derivable Say or Differentiable say it is the same thing said to be derivable or Differentiable et x equi A X E A E V CAN Dr Tange if we can draw a tenth of a line of it Slope of final slope at x = a Twa ect f a on y e f a i.e. if we take a fit slope on x e a if we can make a tensor of that then we will say that the function what is at point is differentiable then Differentiable means you can draw a tense of fin it slope at x = a and I told you mathematically Mathematically, we work out AHD is not the left hand derivative which we call It is said that f ' a minus this becomes the limit a tending to 0 f of a - h - f a up mine a a my mine a and right hand what is the derivative of the right hand side derivative e equal to add of a plus limit extending to zero and a of a p a minus f a its a and The function A is derivable on A then The derivable will be then A is Derivable at x equals a abb when this ld If L hd where at x is equal to a this is equal to Whose turn should this be equal to R HD's eight x equals a e equal to both of them one final it quantity should be equal to that is f d of a my e equal to f d of a py e equal to a fine quantity suppose it comes l then mathematically any function is differentiable is called when its LHDAA is equal to r hd a a and is equal to a fine it quantity means both are fine become equal to the quantity l and this What do we say in the scenario? We say yesss Look at the employees here, both of them How many LD and RD have come, L has come for these Common value is called the common The Value of Common value of l hd end RHD at x e a is called cold derivative of function i called derivative of function at x equals a and e denoted by and how do you denote it and i denoted by FDA and denoted by FDA and y what does it denote and what does it represent Slope Off Tange and It Represents the Slope of Tange at x event a at x ev a i.e. assume l hd and Both of RHD and both of LHD If RHD also gets these two fives then we will say that the value of the derivative of the function There is F on a or we can understand it like this that if I draw a tangent on the graph at x = a then what will be the slope of the same it will become f it will be equal to f And we call it add a equals pa add A equals 5 now some important things here But some important things first The important thing is derivability or differentiable at x equals a differentiable at x equals A Employees t at x equals a continuity at x equals a but its If the reverse is not true then take care of it hold only this way it does not hold the other The way it does not hold the other way means it is like this Continuity does not imply holding is not differentiable, differential is an impla continuity at x e a ok dis Dis Continuity at ex ect A Implies non differentiability x e Two Non Difference liability liability at x equal to but its reverse is also not true That means non difference bill is not implied dis continuity at x equal to a then Differentiation Ability Implies Differentiable Implies Differentiable Employee Difference E A Employee Tendency Bility Employees Tange Impla Tange the probability tene the probability t x = a i.e. if a function is differentiable at a point So that means what do we build at that point can make a tenge finna it of slope but if we take the tangent at a point If you can make it, brother, it doesn't mean that that the function is differentiable there Tell me quickly as an example Can you give any example as an example? ho bhaiya can you tell me some can you tell me n example for it example whatever you can think of yes tell me sir power of x is 1/3 absolutely correct power 1/3 of x in this case we have the vertical at 0 can make a tenge eight but at that point A function is not differentiable vertically Tangent is formed, the function is not derivable I think I understood it then Differentiable Implies Continuity Discontinuity implies non-differentiable derivable differentiable at x e a implies tange ability at x = a then the next thing that What you have to remember is that the limit is tending to a is not a if f is derivable at x = a if f is Derivable at x = a then limit x tending to a f a - a upon x - a This limit will come from the FDA ok that means whatever its limit will be the same as add a it will be ok then the limit extending to a f a minus a my ex mine is this but in this I will not even say that, assume that its limit is L But it would be better if its what is the limit if it is L then whatever is its limit The same add a will come to be better than that Suppose its limit is a, then it will be whatever The value of add A will be the same, i.e. whatever its value The limit will be added as soon as it comes, it is clear because right hand limit right hand The derivative becomes its left hand side The limit becomes the left hand derivative ok then one more important point e l HD End RHD IF L HD&R hd at x e a r fine it are you fine it but Unequal Butt unequal than f Judge How would the continuous at x equal a be? Continuous Hoga and Geometry Imp & Geom Matriculation Employees what does implant do tell me quickly Ematic Impla Sharp Corner EX E sharp Corner at X Equation means if LSD and RSD are final it is unequal if the function is derivable then it will not happen but it will definitely be continuous and What does geometric mean? The sharp corner is at x equals a Ok then va equals a eks is said to have I said to have I said to have a vertical Tange Vertical tange at x equals A If when if both l hd both l hd and r hd at x eq A R plus infinity and minus infinity ie K Agar L HD and RD both together plus infinity or minus infinity goes to a Then we say that what is on A is on A. vertical tense is we will have vertical tange okay wa i f is a this function is set Two have I set to have a cusp cusp et x e a If L hd at x e A is Plus infinity and r hd at x it a is minus infinity ld e i minus infinity then in that case we will have a cusp said to have cusp at x i a if ld x e plus infinity and rd x e e minus Infinity then when will you remember this thing The cusp also has a sharp point so cusp is also sharp point yes vertical In case of Tange In case of Vertical Tange you can say yes there is tenge but the function will not be differentiable The function will not be differentiable I hope this I'm clear too, now one more thing See what the regions of non-differentiable will be region region of non differentiable region of non differentiable what will be the regions now let's see the regions of non differentiable What will be the region of non-differentiability Let us see the region of non differentiable if See if he will be number one Discontinuity First Discontent Gent Vertical Tenge So Dj R There are three basic reasons for If there is a discontinuity then a sharp corner will appear or Will there be a vertical tension or a discontinuity? it will go okay the major regions are the Region Fore cusp cusp is also a sharp corner Cusp is also sharp corner to vertical What is the difference between tange at and cusp If the function has a cusp then at that point what becomes a sharp corner becomes a sharp corner But when the vertical angle is at, there The graph smoothly passes out that at the point such as the origin in the power 1/3 of x but ok This is clear now, then some golden points Some golden points that you should know What is it that you need first yes some golden points And what are the golden points, let's see are you what are those golden points The golden points are something like this From are the most First Even Difference Product of two derivatives of two differentiable functions off to differentiable Functions Of two differentiable functions is differentiable That is, if a function is differentiable and The second function is also differentiable, then both If we add or subtract and multiply, then Whatever resultant function comes, it will Differentiation will be another important factor second important thing second important thing The important thing is this k if f is fg are derivable at x Equal A&G is not equal to Zero then f by x is derivable at x equals a is derivative at x equals a i.e. if two The function is differentiable and not zero then f will be the derivative then the next point If f is derivable then if f is derivable then x equals A&G is non Derivable nonce derivable at x equal to a then f x plus minus g x how is that will be e non derivable at x e I Non Derivable X e i.e. if is a derivable and is a derivable If not, then the new function that will be created by combining both will come that too will not be derivable on A that too Derivable on A will not be correct is a plus minus a is non-derivable X e then the next point if f is derivable and g is non derivable if f is a Derivable &g ex is derivable is derivable at x e a then nothing definite than nothing definite Can Be sadan nothing definitely can be said about late some about there some about there sorry The product and quit about the product and By multiplying both the qubits i.e. Or how will it work by dividing, we don't say anything We will have to investigate as much as we can Check means we cannot say anything without checking anything can happen from we can't anything with Certainty everything has to be checked in this In case of sum and difference then we have We can say in case of sum and difference you can, yes brother limit exists a function not derivable but in this case we can't say anything and this is where our atonicity Here comes our atonicity Atoning result atoning result is coming now what is it atoning result over here and see are atoning results what was atoning The result was if wa i a x equals f a nj a written ho how should this function be continuous yes Continuous at x equals a and this function How about the derivative of x equals a at Derivable at x equals a and plus a of a becomes zero and f a is 0 then we can say is this product an f ex is derivative at x equal to the f x will be derivable on x e a so that was the atoning result here yes sir this was the atoning This is the result, our atoning result okay and in this we have another important The point we know as S.N.C. it was that it was if f x is equal to a x in x a one in g x here if g x what are you if g x S G X S A Fine it is fine But Unequal Butt Unequal L HD&R hd at x equal to A fine it l hd and r hd at x Equate to a and how is this function derivable et x equi a derivative at x equals to a then Pa Ecken B Derivable than and perhaps it was also written tha and e 0 and g equal to 0 then f a x can be Derivable can be Derivable Only if f off a row be f of a how much be row be if not then this cannot be derived i.e. 5x For g to be derivable, a means of 5x to be derivable where g is a function where g should say a one l hd and RSD is fine but it is unequal and g off a how much is row so such a function Derivable at a can only be if a is 0 a is derivable only if a is 0, otherwise it is No You will have the same result, we will see continuity I can also say yes son this is how we are We can also say result for continuity what can you say in continuity it's ok to listen y if you are f a nudge a f a in g a here l HL&R HL R FINNA IT R FINNA IT But unequal are fine it but unequal finna it ho and unequal ho ok And this F, how should it be, should it be continuous How should it be, should it be continuous, okay and yes Its GE should be defined defined is defined and g So this function then fi can be then f x Can B Continuous at x equals A Only if only If a of a equal to 0 then what is the amount of a of a Zero is absolutely true if you want this result You can also apply it in continuity, okay now You will say sir yes son this is the result you can also apply it in continuity, it's ok Like why would this happen as an example and also Look here if I put a limit here tending to zero right hand limit if I see f f of a p h it's of a p h what will be its limit sir Its limit will come F A, assume its right hand limit is g's now limit is a tending Two Zero f of a minus h in g of a minus h it It will again become A of A in A and F of A what will happen s f of a hoga f a in g of A now what should these three be, continuity for sir these three should be equal then a Off a in l e f a in a e f * g of a then if I want these three to be equal, so first So look at this, these two should be equal So I took the f a common what was left inside l - m = 0 but a and m are not equal then f a has to be zero and a of a What will happen to everyone once they reach zero? How will the continuous i.e. F be continuous This is why when we move x into Greatest Integer of X Watch Now Suppose a given one okay and I'm looking at these x Belongs to Mine If you set myview then it will be zero then it will be zero will be the point x of the point continuity Equals 0 F E Continuous by Atoning The result is Na by atoning result continuous at x equals 0 at x equals g stunning result will become continuous from but the rest of it is a function like x by 2 which will have a range Here it would be - 1/2 to 1 - 1/2 to 1 If x = 2 then where will the integer be formed in this Sir when x ba 2 should I take some more friend should I take 4 okay here sir this will happen so in this which will become the integer x = 2 i equal to 1 0 1 and t is equal to x = 0 2 and then This will be its point of discontinuity But how did this function become zero? Sir it became continuous at zero so this It has become continuous, now where can it break Sir on two on two it will be discontinuous and Maximum is coming on four there also If it is discontinuous, then it means It will be discontinuous on two and on four but l hd rhd but lhl and RHL both will be finna it so this is jo The outside function is in the product if I I want this product to be like this This can only happen if the product is continuous is this implies 5x or fx4 will be sir discontinuous et x equals 2 and four because we know sir This is discontinuous from outside It is continuous and sir here on two and four but lhl rhl final it is uneven And this function is also defined on two, on four. If this function is defined then the outside function when does not become zero until this function It cannot be continuous, so the function Non-continuous on two as well as on four How will it be discontinuous on four? hoga it will be discontinuous at the end what should happen on four two and four will become discontinuous if it to But if it was becoming zero then definitely If it becomes continuous then many times children would think yes sir how did you write it directly like this I write after thinking like this, okay understand this If I came in then that means Estonia was different in exactly the same way We can write an ASN like this: You can also write it in continuity but this Only those people will know what people are doing are the people who follow us completely who are also watching these revision lectures Well, I told you somewhere in the class It must be okay brother, I understood this brother Yes sir it is clear very well I understood it, brother, I understood it completely If it would have come, then we would have got exactly the same result You can put it here also, now let's come to the next on whom next we come out we have a look at the mud fxxx.pro How do we check its differentiability in this Let's put A as zero. Two put one equal two 0 he got it from here x equals a b and so on if at x equals a f d x add x becomes zero then mud a x is derivable at x equals a at x equals a else if at x equals a if at x equals a what should happen ad a f d ex is not equal to 0 then md f is equal to none Derivable at one equals is non day at x equals a similar check for another Points Similarly check for x = be and so on then here We consider the mode f to be non-differentiable of one If we are discussing the points then mud f a There is a possibility of non-derivable where f becomes a 0 and where f 0 Suppose it is formed on a, b and c, then the f becomes 0 at the points when finding the mode So when the negative part of the graph is shown upwards When we reflect, there is a sharp There is a possibility of corners becoming sharp A corner is formed when the x axis is perpendicular to that point cuts the curve at that point And sharp corner is not formed when x axis If it touches the curve at that point then This is what we need to find out, whether there the x axis will cut or touch So, we can calculate the value of the derivative there. and if we take it out and how does the derivative look like If the derivative comes out to be positive then the derivative If it reaches zero then it will touch the x axis And how can we say if the derivative is non-zero? If it goes then it will cut the x axis i.e. If you reflect, you will see a sharp corner You will get the receipt there, it is clear I think I understood this well It must have become clear brother, this is very good this must have also become clear from this, yes sir understand this When I came to this, I told this very well This would have been cleared The function mode f of one we call non differentiable If you can take it out from the point then you will We told you the stoning result, tell SNC and along with that we gave that SNC By extending you in continuity as well By proving it, we have told you here I think it is so clear now Let's see if there's anything left brother So now all that is left is this Determination of function then determination of We have already told you about the function. that which is the given functional equation, the function what is the function doing to the relation satisfies her with respect to y In differentiable then set y to zero boundary You can customize your app by using value conditions. find the value of f f d a and Then by integrating it, f will come to one and then you can find out the desired function So that is what it is, it is a mechanical method you can go and get the complete differential You can see the lecture, it is in the last Differentiable Last Two Four Problems If you try from there, it will happen automatically So that's all I'll keep for today, thank you. And that is all from my side as of today So that's it for today and I hope this In the mind session you discussed each and every thing very carefully. You must have revised each topic thoroughly very well mapped out in your mind If yes, then this was the entire LCD time No doubt it felt a bit excessive but Each topic must be revised one by one, then See you in the next mind session, until then Till then Jai to the mind map session Shri Krishna Radhe Radhe