Understanding Limits in Calculus

so respectable we have started the new topic as limit in the last class let me give you a short brief in the last class we discussed about this phenomena what does it mean by say that limit let me give you a 10 minutes short reminder about this we have discussed this limit as if y is equal to FX is a given function so x is going to be independent variable, y is going to be dependent variable. What does this mean? That statement means that if x varies, because y is dependent on x, then definitely y means fx will also vary. Let's take some example like, if I say we have taken a function y is equal to 3x plus 5. If x is equal to 1, So y is going to be 8. This means if x will have a tendency to go towards 1, then y will try to be 8. And when x will be exactly 1, then y will be exactly 8. Also, when this statement is written, so we write this statement as limit x tending to of fx is going to be L. Where two qualities are very important in L. L should be definite and L should be finite. L should be definite and finite. These two qualities are very important in L. If L is definite but not finite. If L is finite but not definite, then also L is of no use. What does it mean? How will it be definite or finite? By trying several enormous questions, I will give you a brief but a solid example of what this statement means exactly. So for now, when we talk about this concept limit, so we would write if independent variable If we go towards a finite given number of x, then our function f of x if a fixed number of f of x goes towards l, if the tendency develops, then we say that the limiting value of this function is l, or limit x is tending to a of f of x is equal to l. Don't think that f of x is equal to l, no. He is saying that if... x a ki taraf jaayega to ye function fx l ki taraf jaane ki tendency rakhega this is not exact value of fx this is tending to value of fx agar x a ki taraf jaayega to fx l ki taraf jaane ki tendency rakhega sir l is known as l is known as limiting value of function And also we write this whole statement like limit extending to A of fx is equal to L. Also we have seen what is right hand limit and what is left hand limit. And when we have to use this right hand left hand limit, we have seen it. When we talk about right hand limit, it is denoted by R H L. And we represent this limit as limit extending to A plus of. fx and when you talk about lhl this lhl is known as left-handed limit and we usually represent this like limit extending to a minus of fx let me give you the brief what does it mean by say that limit extending to a plus of fx and limit extending to a minus of fx so respectable exactly what is happening Suppose this is x-axis and this is some point A. I said x is approaching towards A. x is approaching towards A means definitely x is not A. It will only move towards A when x is not A. It is standing somewhere far from A and moving towards A. So when I tell you that x is approaching to A means x is not equal to A. Means x will move towards A. So there are two options to move. It can be x. already here which is greater value of a and a is standing on greater value and maybe going towards xa. It can also be that x is standing over the values lesser than a. and from here it is moving towards xA so when x is moving towards the greater value of point A means it is moving towards A so we say that we are going to calculate right handed limit RHL means x tending to A plus means x is moving towards A but x was actually standing at the plus value of point A and from here it started moving towards A Or it can also be that x was on the left of a and from there it started moving towards a. So we say that x is approaching towards a but actually x was standing over the lesser value, minus value than a. And from here x has started moving towards a. If this limit suppose is going to be l1 and this limit is going to be l2. Suppose rhl's value is l1 and lhl's value is l2. So if. If L1 is equal to L2, so we would say limit exists for fx. And if L1 is not equal to LL, if I have to remove L1, L2, RHL, LHL in any function, and if after removing RHL, LHL, I saw that RHL, LHL are coming to be different, then we will say that limit exists for that function fx. does not exist. In which function will we get RHL and LHL? How will we know? So, for that concept, for understanding it, the best option is the graph of y is equal to greatest integer function. When you talk about this phenomenal graph, so when you talk about this graph, so this graph going to be something like, in this way, this is y-axis, this is x-axis this is origin and when you plot the graph of integer x so 0 say one k beach my y is equal to zero key line just my one not included one say to give each one of our praying it was not twice equal to one killing this my one include over to not included hamaisha greatest integer function When 2 moves between integers, which is 0 and which is 1, 0 will move between 1 and 1 So it will accept the smaller integer and will always leave the bigger one So 0 and 1 have blank dots on 1 This graph is not suitable for 1, it is only suitable for the value between 0 and 1 y equals 0 When you move between 1 and 2 on x, y is equal to The line of 1 will be this graph but it will not work for 2. When you move between 2 and 3, then go to 2 per close bracket y is equal to 2 height. And 2 height per, this is the line y is equal to 2 which will not work for 3. It will be a graph for 2 to 3 values. Why we are discussing this graph? So respectable this graph seems to be the best option to understand the concept. What is RHL and LHL and moreover How will we know by looking at the function that in this function if at any point we want to calculate limit so we have to take the support of RHL or LHL Suppose I have been given a question limit x tending to 2 of greatest integer x What will the students do? If they don't know the real meaning of limit, then they will take x is equal to 2 and put it in this function. So this will be greatest integer of 2. This is 2. Why? Because by putting a limit, the function is neither 0 by 0 nor infinity by infinity. Rule says if 0 by 0 or infinity by infinity is not there, then the value is the value of limit. But is this correct way? No, not at all. We have to study the graph of greatest integer function in this condition very thoroughly. Definitely this is going to be the wrong calculation, the wrong way to calculate the limit. What's wrong in that? Respectable, when x2 is moving, then we consciously see the graph of greatest integer function, greatest integer x function. When I look at the x-axis, If I move towards 2, then there are two ways of moving towards 2 on x axis. One, I move towards 2 by standing on a higher value than 2. Or I move towards 2 by standing on a lower value than 2. When I try to move towards 2 by standing on a higher value than 2, So, the graph which is going to support me at that time is this part of graph. Means when I am calculating from the right side of 2 to the 2, which is the RHL, then this value of function is supporting me, that is 2. But when I try to move from the left side of 2 to the 2, then the part of graph which will support me, that is y is equal to 1 part. So, this point... I am calculating the limit of the greatest integer function Now please listen to me carefully I repeat my words According to this function The point at which I am calculating the limit of this function Respectable to the right, the graph is something else And the graph is something else on the left Means it is a clear cut indication That you can't directly substitute the limit in this You have to calculate the R and L in this How did you know? graph se, kyunki right se graph kuch aur, left se graph kuch aur, matlab aapko isme calculate karna padega, first of all right hand limit, so right hand limit is defined as limit x tending to 2 plus of greatest integer x, aur ye main graph se hi dekh sakta hu, jen mein 2 plus par khana ho ke 2 ki taraf doorunga, to function kya value dega, 2, straight away iski RHL is going to be 2, but when you calculate LHL, left handed limit This is limit extending to 2 minus of greatest integer x So when I will stand on a value less than 2 and turn towards 2 So the value of the graph that will support me that is going to be 1 So for 2, RHL for this function is going to be 2 For 2, LHL of this function is going to be 1 Since RHL is not equal to LHL I would say limit does not exist for this function at x is equal to 2. Listen carefully. Amazing information. The limit of function at x is equal to 2 does not exist. I will say L1. Suppose this part is L1 and this part is L2. So, respectable L1 is not equal to L2. So, limit x tending to 2 of greatest integer x Does not exist, is it? Definitely? Does not exist. Look at this amazing thing. What is the phenomenal thing hidden in this? If I talk about limit x is tending to 2, then definitely limit does not exist. Does function exist on x is equal to 2? This function exists on x is equal to 2. So even if limit does not exist, But the meaning of limit exists is not that the function will not exist at that point. It is possible that the function exists at the point at which you are taking out the limit, but the limit of that function does not exist at that point. It is also possible that the limit of the function exists at that point, but the function does not exist at that point. So if ever rHL is not equal to LHL means limit does not exist and if someone asks you that if at any point, any function's limit does not exist according to rHL and LHL then will it happen that function itself does not exist at that point? it may happen but it is not mandatory that if limit does not exist then function also will not exist. See on 2 limit does not exist but does function exist? It does. So remember that RLHL and existence of function at that point is at all a different concept It may be possible that we may get an example that if RLHL and RLHL do not exist Function also does not exist at that point But if RLHL and RLHL do not exist Then we cannot say that confidently that function also does not exist at that point See function does exist at x equal to but limit does not exist exist. In the next limit we will try some more good question concepts. So respectable let's start doing some methods of evaluation of limit means question based on methods of evaluation of limit we discussed in the last class that how many methods are there to evaluate a limit first of all we have got these few questions based upon the very first method that is direct substitution method after this we saw factorization method then we saw some standard limit, vesselization method then we L-Hospital method, series expansion and then some standard limit like 1 power infinity. So, respectable, let's begin today's lecture by solving these kind of very basic questions. We will try to take it to the highest level by starting with very basic questions. So, this is going to be the very first question based upon the very first method, direct substitution method. Let me read the statement. Question is given to me. We want to evaluate limit X tending to 1. 3x squared plus 4x plus 5. Now please listen to me carefully. This is a very small thing but very important. From basic level to IET level, I repeat my words. Listen to me carefully. Respectable. From basic level to IET level, as soon as I get any question of limit, first of all, Make a habit to take the value of limit directly and put it in the function. As I put x as 1, this is 3 into 1 square plus 4 into 1 plus 5. And this is going to be 3 plus 4 plus 5. This is going to be 12. Is this 0 by 0 or infinity by infinity? No. This is going to be a very sweet, well-known value. in the context of number system. This is a known number, 12. So, if ever in the question of limit, by putting a straight limit, 0 by 0 or infinity by infinity, is not coming. That means, you can put a straight limit in that question. And the value that will come, that will be the value of limit. Let's see one more question. Limit x tending to 2. I have been given x square minus 4 upon 3 plus x. First of all, Whatever the question of limit is, I will directly take this value and here I will substitute. Put x is equal to 2. And if really I have done so, this is going to be limit x tending to 2 of 4 minus 4 upon 3 plus 2. This is going to be 3 plus 2. So this is going to be 0 upon 5. Is it 0 by 0? No. Is it infinity by infinity? 0 upon something is going to be 0. So value of this limit is going to be 0. I have not done any solution. I have directly substituted the value of limit. If this is not giving me indeterminate form like 0 by 0 or infinity by infinity. So respectable we are permissible to substitute x is equal to 1 directly in the equation. That is called direct substitution. Let's check it out how this limit will be calculated. Let me read the statement limit x tending to 0. We want to calculate this limit root over 1 plus x plus root over 1 minus x whole upon 1 plus x. As soon as I get the question of limit, first of all I will take x is equal to 0 and I will substitute. This value in the limit and I will check it out. Kya mujhe infinity by infinity as 0 by 0 ka form mil raha hai? Respectable no sooner we have substituted x is equal to 0. This is root over 1 plus 0. Root over 1 minus 0 whole upon 1 plus 0. This is 1 plus 1. What? 2 upon 1. This is going to be 2. Is it 0 by 0? No. Is it infinity by infinity? No. So this will be. the value of this limit so how the direct substitution work these are going to be few example how do we treat the question when direct substitution method is involved or cup direct substitution possible job c the limit darkness is 0 by 0 yeah infinity by infinity next time question try to think 0 by 0 or infinity by infinity So respectable here it falls. The next method the question based upon next process factorization method. Here are few example in which factorization method would be. used. Jaise hi limit ka koi bhi question mujhe diya jayega. First of all, main sebe limit uthaunga aur function mein plugin karunga and definitely I will estimate what result is coming out to be. X is equal to 2, main direct rakhaa. Dekhe jaise rakhinge 4 plus 6 is 10, 5 2s are 10, this is going to be 0. And denominator also is going to be 0, this is going to be 0 by 0 form. Which means abhi main. This function can't put limit x is equal to 2. Then what should be the plan? I have seen the function carefully. This is a polynomial function. Polynomial function means there is a chance of factorization method. Let's see. Let's begin the solution. Let's see what happens if we factorize this. And if really we have done so. Obviously, this will be factorized. x minus 2 into x minus 3. This will be factorized. a plus b into a minus b as 4 is 2 square this is a square minus b square 4 this is x minus 2 as a common factor got cancelled so limit x is tending to 2 it is x minus 3 upon x plus 2 or finally as soon as you cancelled the common factor p Similarly, I am permissible to put limit when I put x is equal to 2 in the function. This is 2 minus 3 upon 2 plus 2. This is going to be minus 1 upon x to 2. So, as I put x to 2, this is going to be 2 plus 2. This is going to be minus 1 upon 4. So, limiting value of this function is minus 1 upon 4. Let's pick the second question. This is limit x is tending to 1 of x cube minus 1 upon x minus 1. As soon as I got the limit problem, I did a substitution with c. x is equal to 1. And when I put x is equal to 1 in the function, so numerator is going to be 0 as well as denominator is going to be 0. Function is giving me the form 0 by 0 which is indeterminate form. As soon as I put a limit in the function, I will get indeterminate form. That is a clear cut indication. We cannot put a limit in this question. Then what should be the plan? This upper part, I am finding it being a form of a cube minus b cube. This is a minus b into a square plus ab plus b square and denominator is going to be x minus 1. And if I cancel out the common factor, I will remain with limit x tending to 1. This is x square plus x plus 1. And finally, as soon as x goes, 1 put here, this is going to be 1 square plus 1 plus 1, that is 3. So limiting value of this expression for this limit is going to be 3. So remember whenever you want to put a limit, the question is to put the value of x directly Check if it is 0 by 0 or infinity by infinity If not, then the value is the acceptable value If it is 0 by 0 or infinity by infinity, that means you cannot substitute the value of x Let's see the next question, how this question will be solved, what should be the idea I saw the question of limit, first come first, I will be taking this value and I will substitute this value in numerator as well as in the denominator. You can see by putting, x is equal to 2, this is 8 plus 4, 12. 2 square 4 into 3, 12, 12 minus 12, which is 0. If you plug in below, immediately you will find denominator also going to be 0. So as soon as I put x is equal to 2 in the expression, then above also 0. Below is also 0. This is 0 by 0 in determinate form. That is a clear cut indication for me. Right now I cannot substitute the limit. Then what option we are left with? Definitely polynomial expression. There are chances of factorization. How to factorize? This is cubic polynomial. So respectable right from this question. Question seems to be a basic one. But definitely it is a basic question. You will get to know a very good thing today. What will be the plan? What is the plan? x is equal to 2 As soon as I put it in this polynomial In which? x cube minus 3x square plus 4 So this polynomial is 0 What does it mean? It means x minus 2 will be our factor of this expression. And if I know that x minus 2 is its factor, then to get the rest of the factors, I will divide this whole expression from x minus 2. This is x cube minus 3x square plus 4. And for this, there is a whole long division process. Definitely, I am not going to do it. You know how to solve it with a long division process. I will let you know about the... Phenomenal approach, when ever cubic expression is given and you have to divide it by linear factor in a second, in a fraction of second, how can you do this work? If you proceed the whole process using the long division phenomena, definitely it is going to kill your time. So what is the short method? If you have to divide cubic polynomial by linear factor, then how can we do this work in a second? So, the process is long and you know that you are not going to do this. I have confirmed that by adding x is equal to 2, it will be 0. It means that it will have a factor of x minus 2. I got one linear factor. It means that two more factors will be hidden in it because it is cubic. I got only one. I need two more factors. So, instead of using long region process, let's do this work by adopting this smart approach. What is the smart approach? The number from which it became 0. Write this number like this. And the rest of the cubic polynomial which I have to factorize. Write its coefficient. X cube's coefficient is 1. Square's coefficient is minus 3. Then X has no coefficient. It is 0. Constant term is 4. I have written coefficient. After that, look carefully. I have written this number as it is here. This will happen. multiply this number by 2 write the result here and add both of them what will you get? minus 1 write the result here and add both of them minus 2 this number will be multiplied by this this is minus 4 always add both of them this is 0 so cubic was divided by linear so the result will be this is the coefficient of x square This is the coefficient of x, this is the constant and this is going to be the remainder. So what will be the remainder? It will be 0 because you have divided it by a factor. So immediately I got that if I divide x cube minus 3x square plus 4 by x minus 2, so this is going to be what? x square minus x minus 2. So definitely x cube minus 3x square plus 4 can be expressed as x minus 2. into x square minus x minus 2. So, I got its factorization in this form. So, respectable, let's see what happens if we go through proceeding this in this way. And if really we have done so, this limit is going to be, limit extending to 2, this was x cube minus 3x square plus 4 whole upon x power 4 minus 8x square plus 16. So this part of mine now has become, after getting this part factorized, this is limit extending to 2, what will we get? x minus 2. How will this part break? x square minus x minus 2 whole upon x power 4 minus 8x square plus 16 if I am not wrong. And if you focus upon this whole part, this is limit extending to 2, x minus 2. This will be factorized as x minus 2 into x plus 1 whole upon. Look carefully at this part. What arguments will be there? Here it is written x square of square. Last term it is written as 4 square. Whenever first or last term if it is perfect square, then understand either it will be a plus b whole square or it will be a minus b whole square. Take a chance and see. No, we will factorize it from splitting the middle term. This term is also perfect square and last term is also perfect square. So, let me see the term of the middle. It is written as x square square. It is written as 4 square. So, can you write the term of the middle like this? 2 into 4 into x square. Definitely. So, this is the form of a square plus b square minus 2 ab. So, very obviously and very smartly. Let's talk about the upper part. This is x minus 2 whole square into x plus 1. and if we look at the lower part, then this is whole square of x square minus 4 again this is a square minus b square Form can be factorized as a plus b into a minus b. So, respectively this part of limit goes off. Limit extending to 2. Upare x minus 2 ka whole square into x plus 1 whole upon. This is x plus 2 into x minus 2 getting whole square if I am not wrong. And if you solve this part further. So, this limit of mine goes off. x minus 2 whole square into x plus 1 whole upon this is x plus 2 square square into x minus 2 square common factor cancel and as soon as common factor cancel from top to bottom then this factor gets cancelled due to which this limit becomes 0 by 0 or infinity by infinity but they did just say This common factor is cancelled from top to bottom. So this is the problematic part due to which you were not able to put the limit in the limit. And after this as x approaches to 2, wherever x is 2, 2 plus 1, this is 3, 2 plus 2, this is 4. So limiting value of this function is going to be, sorry this is square, square it, square of 4. So this is going to be 3. So value of this limit would be 3 by 60. So these are few questions based upon factorization method. Next question par move karte hai. So respectable let's move ahead on to the third kind of method of solving the limit that is rationalization method. Kya special cheez hai? Whenever you write the terms of the root in the question of limit, that is a clear cut indication, in most of the cases you will have to do the rationalization. Why am I saying this? Most of the cases because when we will be discussing infinite limit, even after the root, we are conscious whether we have to do the rationalization or not. Sometimes in infinite limit, even without rationalization, our limit gets solved. If we look at the basic limit, then if I am given the limit, the terms of root are used, definitely rationalization method seems to be optional. Let's pick the first question, limit x tending to 0, root of 2 plus x minus root 2 whole upon x. As per my habit, as soon as I get a question of any limit, so respectable, I will take the value of x as 0 or whatever limit is there. And I will directly substitute in this question and see what form I am getting. For example, you will put 0 in x. Root 2 minus root 2 is 0. If you put 0 below, it is 0. So overall this is going to be 0 by 0 form. And that gives me the hint. You cannot substitute limit right now in the question. Then what should be the plan? Root is visible. We have to solve the limit. Let's begin the solution. Rationalization seems to be the best option. root over 2 plus x minus root 2 whole upon x. I multiplied numerator by its opposite sign, which means root of 2 plus x plus root 2. And for making it balanced, I divided by this number, root of 2 plus x plus root 2. And freely we have done so. Let's see what result we are going to receive. This is a minus b into a plus b. Limit extending to 0. Do not forget it in writing. This is limit extending to 0. a minus b into a plus b. This is a square minus. This is b ka square whole upon. This is x times what? Root over 2 plus x plus root over 2. Square se root kata. Square se root kata. So this is going to be. Limit x tending to 0, 2 plus x minus 2 whole upon x into root over 2 plus x plus root 2. This 2 and this 2 are cut, so we are left with x upon this part. And when you cancel x from x, so we are going to have limit x tending to 0. This is 1 upon root over 2 plus x plus root 2. And finally... After doing the rationalization, we will replace x directly with the required limit in the part which I have. So we have replaced x as 0. And as I have replaced x with 0, this is 1 upon 2 plus 0, this is root of x. 2 root 2 this is root 2 plus root 2 is going to be 1 upon 2 root 2 so respectable value of this limit is going to be 1 upon 2 root 2 next question let's see this is going to be limit x tending to 0 x upon root over a plus x minus root over a minus x as soon as any question of limit will respectable first of all I will directly substitute x is equal to given limit that is that is x is equal to zero put zero above also if you put zero below then this is going to be zero so question is giving to me as 0 by 0 means in determinate form and if I get in determinate form by putting limit in question that is a clear cut indication you cannot permissible to substitute the limit what option we are left with I can see the terms of root Means rationalization seems to be the best option Let's see, I just rationalized the numerator Now I will rationalize the denominator But if really you will do so Let's see what result we are going to receive So what will be the rationalizing factor of this? Change the middle sign This is root over a plus x plus root over a minus x whole upon This is x into Root over a plus x minus root over a minus x Getting multiplied with root over a plus x plus root over a minus x Now this is limit extending to 0 x into what? Root over a plus x plus root over a minus x whole upon This becomes a minus b into a plus b what? x times, sorry this is not, this is not x, look at this, this is not x, only above x, below x is not there, take care. So this is going to be a minus b into a plus b, this is a square. minus b square. This and this root, this and this root will get cancelled out. This is limit x tending to 0. x times what? Root over a plus x plus root over a minus x whole upon a plus x minus a plus x. a and a will get cancelled out. x plus x is going to be 2x. So this is going to be limit x tending to 0 x times root over a plus x plus root over a minus x whole upon 2x. x and x got cancelled. And finally, the remaining part, if I substitute x with 0 in that part, so as you substitute 0 as x tending to 0, so this root a plus root a, what will you get? 2 root a whole upon 2. cancellation of this will give you root a so limiting value of this limit is going to be root a in the same way third question i can leave for you if you put limit x is equal to 0 see this function this is a minus a 0 below also 0 do reslization after reslization cancel common factor in this way you can easily get the value of this limit So, respectable is a very small thing to see but definitely this whole process, this whole phenomenon is going to take a very important role while solving the question of limits. So, today we saw what does it mean by say that limit, what is direct substitution, when to remove RHL and LHL, what is factorization method and exactly what is rationalization method. Let's move to the next question. So respectable here it falls. Some more questions, few more questions. Very nice question based upon factorization method. This is the first question. Limit x tending to 4. x square minus 16 upon root over x square plus 9 minus 5. And this is second question. Limit x tending to a. Root over a plus 2x minus under root 3x whole upon root over. 3 plus x minus 2 root x. Let's first see how to solve this question. Let's begin the solution for this question. As soon as I get the question of limit, first of all, I will put x is equal to 4 directly in the limit and I will check it out. Do I get 0 by 0 form? Remember, it is a small habit to see, but in every question, put a limit once and check whether it is 0 by 0 form or not. I put 16 minus 16, 0. In denominator, I checked 4, 16 plus 9, 25. 5 minus 5, 0. All in all, this form is going to be 0 by 0 form. Indeterminate form. Means, you cannot use limit in this function. Then what should be the plan? I have seen carefully. I can see a term of root in this. Definitely, I have a hint that I should rest this question. And if really we have done so, let's see. What result we are going to receive? How did I rationalize? I multiplied the part of the lower part by the factor opposite to the upper part. So this is x square plus 9 plus 5 whole upon this is root over x square plus 9 minus 5 multiplying with x square plus 9 plus 5 If I look at the bottom, this is a minus b into a plus b going to be this is limit x tending to 4 do not forget it limit x tending to 4 this is x square minus 16 into root over x square plus 9 plus 5 whole upon this is a square minus b square means root over x square plus 9 whole square minus 5 whole square And when you solve it further, limit extending to 4, this is x square minus 16 into root of x square plus 9 plus 5 whole upon this root and this will cancel out. This is x square plus 9 minus 25. 9 minus 25 is minus 16. So this is limit extending to 4. x square minus 16 into this part. root over x square plus 9 plus 5 and this whole portion is getting divided by x square minus 16 this and this common factor got cancelled or just say you can meet us a common factor cancel we are permissible to put the limit directly in the question and if really we have done so if x tending to 4 just see 4 telling you 4 square plus 9 you 25, 16 plus 9, 25's root 5, 5 plus 5, 10. So value of this limit is going to be 10. In this part, I put 4 to x. So this 4 square, 16 plus 9, 25, 25's root 5 plus 5, 10. So value of this limit is going to be 10. What should be the plan for this? Just like in the question, I saw the limit, so I will first substitute the limit You can see by adding the limit will be 0 by 0 form Definitely we cannot substitute the limit directly What to do? Because I see the root in the numerator and the root in the denominator So I will have to rationalize both Numerator as well as denominator And if really we have done so Let's see what result we are going to receive First let's talk about the numerator, this is like this root over 3x so what will be the rationalizing factor of this? root over a plus 2x plus root 3x to balance this I have to divide it also, let's do it now before that what is the part of denominator? this is root over 3a plus x minus 2 under root x so what do I need to do the rationalization of this? this is root over 3a plus x plus 2 under root x. To balance this, we have to multiply it above. Let's do it. Now this is going to be, what will be its balancing factor above? root over 3a plus x plus 2 under root x. And its balancing factor, which I want below, what will be it? This is root over a plus 2x plus root over 3x. Further, you work irrespective. This is a minus b into a plus b ka form. This is going to be a square minus b square. a square minus b square. This b square into, the remaining part, what is remaining? This one. These two will be adjusted. a minus b into a plus b. Into what is remaining? This is 3a plus x plus 2 under root x. And if we come down and see, then this is a minus b into a plus b. What will happen? Again this is a square minus b square. This is 4x, b square 4x into this part. These two are adjusted here. So this is root over this a plus 2x plus root over 3x. Now 2x minus 3x is going to be this is a so limit x is tending to a 2x minus 3x is a minus x. See below 3x plus x minus 4x x minus 4x minus 3x 3x minus 3x from which you will get 3 out of it. But you got a minus 6. into, there is a part left above, what is that? This is root over 3a plus x plus 2 under root x. And there is a part left below, this is root over a plus 2x plus under root of 3x. Common factor is cancelled and as soon as common factor is cancelled from above and below, we are permissible to put down the limit directly. Wherever there is x, we put a there. And write when x tending to a, what will you get? 3a plus a, 4a. This is 2 under root a. Plus this will also be 2 under root a. Come down. x squared a plus 2a. This is 3a under root 3a. Plus this is also under root 3a. Up here I will get this is 4 under root a. And I will get 2 times under root 3a. Root will be cut from 2. 4 will be cut. So this is going to be 2 upon under root 3. So 2 upon. And see carefully, 3 is also going together. Don't forget this. This 1 by 3 is also going together. So this is also going to be multiplied with 1 by 3. So cut from 2. 2 times. So this 3 is remaining below. This is 2 upon 3 root 3. So this is going to be 2 upon. 3 root 3 so value of this limit is going to be 2 upon 3 root 3 so what is the special thing if you want to see the root of both numerator and denominator without fear apply the process of factorization for both when common factor is cut immediately put the limit and this way you can evaluate the limit based upon factorization method so we have seen today Direct substitution method, factorization method and rationalization method. In the next class we will be discussing about the question, question based upon some standard limit and infinite limit.

L should be definite and L should be finite. L should be definite and finite. These two qualities are very important in L.

If L is definite but not finite. If L is finite but not definite, then also L is of no use. What does it mean? How will it be definite or finite? By trying several enormous questions, I will give you a brief but a solid example of what this statement means exactly.

So for now, when we talk about this concept limit, so we would write if independent variable If we go towards a finite given number of x, then our function f of x if a fixed number of f of x goes towards l, if the tendency develops, then we say that the limiting value of this function is l, or limit x is tending to a of f of x is equal to l. Don't think that f of x is equal to l, no. He is saying that if... x a ki taraf jaayega to ye function fx l ki taraf jaane ki tendency rakhega this is not exact value of fx this is tending to value of fx agar x a ki taraf jaayega to fx l ki taraf jaane ki tendency rakhega sir l is known as l is known as limiting value of function And also we write this whole statement like limit extending to A of fx is equal to L.

Also we have seen what is right hand limit and what is left hand limit. And when we have to use this right hand left hand limit, we have seen it. When we talk about right hand limit, it is denoted by R H L. And we represent this limit as limit extending to A plus of. fx and when you talk about lhl this lhl is known as left-handed limit and we usually represent this like limit extending to a minus of fx let me give you the brief what does it mean by say that limit extending to a plus of fx and limit extending to a minus of fx so respectable exactly what is happening Suppose this is x-axis and this is some point A.

I said x is approaching towards A. x is approaching towards A means definitely x is not A. It will only move towards A when x is not A. It is standing somewhere far from A and moving towards A.

So when I tell you that x is approaching to A means x is not equal to A. Means x will move towards A. So there are two options to move.

It can be x. already here which is greater value of a and a is standing on greater value and maybe going towards xa. It can also be that x is standing over the values lesser than a. and from here it is moving towards xA so when x is moving towards the greater value of point A means it is moving towards A so we say that we are going to calculate right handed limit RHL means x tending to A plus means x is moving towards A but x was actually standing at the plus value of point A and from here it started moving towards A Or it can also be that x was on the left of a and from there it started moving towards a.

So we say that x is approaching towards a but actually x was standing over the lesser value, minus value than a. And from here x has started moving towards a. If this limit suppose is going to be l1 and this limit is going to be l2. Suppose rhl's value is l1 and lhl's value is l2.

So if. If L1 is equal to L2, so we would say limit exists for fx. And if L1 is not equal to LL, if I have to remove L1, L2, RHL, LHL in any function, and if after removing RHL, LHL, I saw that RHL, LHL are coming to be different, then we will say that limit exists for that function fx. does not exist. In which function will we get RHL and LHL?

How will we know? So, for that concept, for understanding it, the best option is the graph of y is equal to greatest integer function. When you talk about this phenomenal graph, so when you talk about this graph, so this graph going to be something like, in this way, this is y-axis, this is x-axis this is origin and when you plot the graph of integer x so 0 say one k beach my y is equal to zero key line just my one not included one say to give each one of our praying it was not twice equal to one killing this my one include over to not included hamaisha greatest integer function When 2 moves between integers, which is 0 and which is 1, 0 will move between 1 and 1 So it will accept the smaller integer and will always leave the bigger one So 0 and 1 have blank dots on 1 This graph is not suitable for 1, it is only suitable for the value between 0 and 1 y equals 0 When you move between 1 and 2 on x, y is equal to The line of 1 will be this graph but it will not work for 2. When you move between 2 and 3, then go to 2 per close bracket y is equal to 2 height. And 2 height per, this is the line y is equal to 2 which will not work for 3. It will be a graph for 2 to 3 values. Why we are discussing this graph?

So respectable this graph seems to be the best option to understand the concept. What is RHL and LHL and moreover How will we know by looking at the function that in this function if at any point we want to calculate limit so we have to take the support of RHL or LHL Suppose I have been given a question limit x tending to 2 of greatest integer x What will the students do? If they don't know the real meaning of limit, then they will take x is equal to 2 and put it in this function. So this will be greatest integer of 2. This is 2. Why? Because by putting a limit, the function is neither 0 by 0 nor infinity by infinity.

Rule says if 0 by 0 or infinity by infinity is not there, then the value is the value of limit. But is this correct way? No, not at all. We have to study the graph of greatest integer function in this condition very thoroughly. Definitely this is going to be the wrong calculation, the wrong way to calculate the limit.

What's wrong in that? Respectable, when x2 is moving, then we consciously see the graph of greatest integer function, greatest integer x function. When I look at the x-axis, If I move towards 2, then there are two ways of moving towards 2 on x axis.

One, I move towards 2 by standing on a higher value than 2. Or I move towards 2 by standing on a lower value than 2. When I try to move towards 2 by standing on a higher value than 2, So, the graph which is going to support me at that time is this part of graph. Means when I am calculating from the right side of 2 to the 2, which is the RHL, then this value of function is supporting me, that is 2. But when I try to move from the left side of 2 to the 2, then the part of graph which will support me, that is y is equal to 1 part. So, this point... I am calculating the limit of the greatest integer function Now please listen to me carefully I repeat my words According to this function The point at which I am calculating the limit of this function Respectable to the right, the graph is something else And the graph is something else on the left Means it is a clear cut indication That you can't directly substitute the limit in this You have to calculate the R and L in this How did you know? graph se, kyunki right se graph kuch aur, left se graph kuch aur, matlab aapko isme calculate karna padega, first of all right hand limit, so right hand limit is defined as limit x tending to 2 plus of greatest integer x, aur ye main graph se hi dekh sakta hu, jen mein 2 plus par khana ho ke 2 ki taraf doorunga, to function kya value dega, 2, straight away iski RHL is going to be 2, but when you calculate LHL, left handed limit This is limit extending to 2 minus of greatest integer x So when I will stand on a value less than 2 and turn towards 2 So the value of the graph that will support me that is going to be 1 So for 2, RHL for this function is going to be 2 For 2, LHL of this function is going to be 1 Since RHL is not equal to LHL I would say limit does not exist for this function at x is equal to 2. Listen carefully.

Amazing information. The limit of function at x is equal to 2 does not exist. I will say L1.

Suppose this part is L1 and this part is L2. So, respectable L1 is not equal to L2. So, limit x tending to 2 of greatest integer x Does not exist, is it? Definitely? Does not exist.

Look at this amazing thing. What is the phenomenal thing hidden in this? If I talk about limit x is tending to 2, then definitely limit does not exist.

Does function exist on x is equal to 2? This function exists on x is equal to 2. So even if limit does not exist, But the meaning of limit exists is not that the function will not exist at that point. It is possible that the function exists at the point at which you are taking out the limit, but the limit of that function does not exist at that point.

It is also possible that the limit of the function exists at that point, but the function does not exist at that point. So if ever rHL is not equal to LHL means limit does not exist and if someone asks you that if at any point, any function's limit does not exist according to rHL and LHL then will it happen that function itself does not exist at that point? it may happen but it is not mandatory that if limit does not exist then function also will not exist.

See on 2 limit does not exist but does function exist? It does. So remember that RLHL and existence of function at that point is at all a different concept It may be possible that we may get an example that if RLHL and RLHL do not exist Function also does not exist at that point But if RLHL and RLHL do not exist Then we cannot say that confidently that function also does not exist at that point See function does exist at x equal to but limit does not exist exist.

In the next limit we will try some more good question concepts. So respectable let's start doing some methods of evaluation of limit means question based on methods of evaluation of limit we discussed in the last class that how many methods are there to evaluate a limit first of all we have got these few questions based upon the very first method that is direct substitution method after this we saw factorization method then we saw some standard limit, vesselization method then we L-Hospital method, series expansion and then some standard limit like 1 power infinity. So, respectable, let's begin today's lecture by solving these kind of very basic questions.

We will try to take it to the highest level by starting with very basic questions. So, this is going to be the very first question based upon the very first method, direct substitution method. Let me read the statement. Question is given to me. We want to evaluate limit X tending to 1. 3x squared plus 4x plus 5. Now please listen to me carefully.

This is a very small thing but very important. From basic level to IET level, I repeat my words. Listen to me carefully. Respectable. From basic level to IET level, as soon as I get any question of limit, first of all, Make a habit to take the value of limit directly and put it in the function.

As I put x as 1, this is 3 into 1 square plus 4 into 1 plus 5. And this is going to be 3 plus 4 plus 5. This is going to be 12. Is this 0 by 0 or infinity by infinity? No. This is going to be a very sweet, well-known value. in the context of number system.

This is a known number, 12. So, if ever in the question of limit, by putting a straight limit, 0 by 0 or infinity by infinity, is not coming. That means, you can put a straight limit in that question. And the value that will come, that will be the value of limit.

Let's see one more question. Limit x tending to 2. I have been given x square minus 4 upon 3 plus x. First of all, Whatever the question of limit is, I will directly take this value and here I will substitute. Put x is equal to 2. And if really I have done so, this is going to be limit x tending to 2 of 4 minus 4 upon 3 plus 2. This is going to be 3 plus 2. So this is going to be 0 upon 5. Is it 0 by 0?

No. Is it infinity by infinity? 0 upon something is going to be 0. So value of this limit is going to be 0. I have not done any solution. I have directly substituted the value of limit. If this is not giving me indeterminate form like 0 by 0 or infinity by infinity.

So respectable we are permissible to substitute x is equal to 1 directly in the equation. That is called direct substitution. Let's check it out how this limit will be calculated. Let me read the statement limit x tending to 0. We want to calculate this limit root over 1 plus x plus root over 1 minus x whole upon 1 plus x. As soon as I get the question of limit, first of all I will take x is equal to 0 and I will substitute.

This value in the limit and I will check it out. Kya mujhe infinity by infinity as 0 by 0 ka form mil raha hai? Respectable no sooner we have substituted x is equal to 0. This is root over 1 plus 0. Root over 1 minus 0 whole upon 1 plus 0. This is 1 plus 1. What? 2 upon 1. This is going to be 2. Is it 0 by 0?

No. Is it infinity by infinity? No. So this will be. the value of this limit so how the direct substitution work these are going to be few example how do we treat the question when direct substitution method is involved or cup direct substitution possible job c the limit darkness is 0 by 0 yeah infinity by infinity next time question try to think 0 by 0 or infinity by infinity So respectable here it falls.

The next method the question based upon next process factorization method. Here are few example in which factorization method would be. used.

Jaise hi limit ka koi bhi question mujhe diya jayega. First of all, main sebe limit uthaunga aur function mein plugin karunga and definitely I will estimate what result is coming out to be. X is equal to 2, main direct rakhaa.

Dekhe jaise rakhinge 4 plus 6 is 10, 5 2s are 10, this is going to be 0. And denominator also is going to be 0, this is going to be 0 by 0 form. Which means abhi main. This function can't put limit x is equal to 2. Then what should be the plan?

I have seen the function carefully. This is a polynomial function. Polynomial function means there is a chance of factorization method.

Let's see. Let's begin the solution. Let's see what happens if we factorize this. And if really we have done so.

Obviously, this will be factorized. x minus 2 into x minus 3. This will be factorized. a plus b into a minus b as 4 is 2 square this is a square minus b square 4 this is x minus 2 as a common factor got cancelled so limit x is tending to 2 it is x minus 3 upon x plus 2 or finally as soon as you cancelled the common factor p Similarly, I am permissible to put limit when I put x is equal to 2 in the function. This is 2 minus 3 upon 2 plus 2. This is going to be minus 1 upon x to 2. So, as I put x to 2, this is going to be 2 plus 2. This is going to be minus 1 upon 4. So, limiting value of this function is minus 1 upon 4. Let's pick the second question.

This is limit x is tending to 1 of x cube minus 1 upon x minus 1. As soon as I got the limit problem, I did a substitution with c. x is equal to 1. And when I put x is equal to 1 in the function, so numerator is going to be 0 as well as denominator is going to be 0. Function is giving me the form 0 by 0 which is indeterminate form. As soon as I put a limit in the function, I will get indeterminate form.

That is a clear cut indication. We cannot put a limit in this question. Then what should be the plan?

This upper part, I am finding it being a form of a cube minus b cube. This is a minus b into a square plus ab plus b square and denominator is going to be x minus 1. And if I cancel out the common factor, I will remain with limit x tending to 1. This is x square plus x plus 1. And finally, as soon as x goes, 1 put here, this is going to be 1 square plus 1 plus 1, that is 3. So limiting value of this expression for this limit is going to be 3. So remember whenever you want to put a limit, the question is to put the value of x directly Check if it is 0 by 0 or infinity by infinity If not, then the value is the acceptable value If it is 0 by 0 or infinity by infinity, that means you cannot substitute the value of x Let's see the next question, how this question will be solved, what should be the idea I saw the question of limit, first come first, I will be taking this value and I will substitute this value in numerator as well as in the denominator. You can see by putting, x is equal to 2, this is 8 plus 4, 12. 2 square 4 into 3, 12, 12 minus 12, which is 0. If you plug in below, immediately you will find denominator also going to be 0. So as soon as I put x is equal to 2 in the expression, then above also 0. Below is also 0. This is 0 by 0 in determinate form. That is a clear cut indication for me.

Right now I cannot substitute the limit. Then what option we are left with? Definitely polynomial expression. There are chances of factorization.

How to factorize? This is cubic polynomial. So respectable right from this question.

Question seems to be a basic one. But definitely it is a basic question. You will get to know a very good thing today. What will be the plan?

What is the plan? x is equal to 2 As soon as I put it in this polynomial In which? x cube minus 3x square plus 4 So this polynomial is 0 What does it mean? It means x minus 2 will be our factor of this expression.

And if I know that x minus 2 is its factor, then to get the rest of the factors, I will divide this whole expression from x minus 2. This is x cube minus 3x square plus 4. And for this, there is a whole long division process. Definitely, I am not going to do it. You know how to solve it with a long division process. I will let you know about the...

Phenomenal approach, when ever cubic expression is given and you have to divide it by linear factor in a second, in a fraction of second, how can you do this work? If you proceed the whole process using the long division phenomena, definitely it is going to kill your time. So what is the short method? If you have to divide cubic polynomial by linear factor, then how can we do this work in a second? So, the process is long and you know that you are not going to do this.

I have confirmed that by adding x is equal to 2, it will be 0. It means that it will have a factor of x minus 2. I got one linear factor. It means that two more factors will be hidden in it because it is cubic. I got only one. I need two more factors. So, instead of using long region process, let's do this work by adopting this smart approach.

What is the smart approach? The number from which it became 0. Write this number like this. And the rest of the cubic polynomial which I have to factorize.

Write its coefficient. X cube's coefficient is 1. Square's coefficient is minus 3. Then X has no coefficient. It is 0. Constant term is 4. I have written coefficient. After that, look carefully.

I have written this number as it is here. This will happen. multiply this number by 2 write the result here and add both of them what will you get? minus 1 write the result here and add both of them minus 2 this number will be multiplied by this this is minus 4 always add both of them this is 0 so cubic was divided by linear so the result will be this is the coefficient of x square This is the coefficient of x, this is the constant and this is going to be the remainder. So what will be the remainder?

It will be 0 because you have divided it by a factor. So immediately I got that if I divide x cube minus 3x square plus 4 by x minus 2, so this is going to be what? x square minus x minus 2. So definitely x cube minus 3x square plus 4 can be expressed as x minus 2. into x square minus x minus 2. So, I got its factorization in this form. So, respectable, let's see what happens if we go through proceeding this in this way. And if really we have done so, this limit is going to be, limit extending to 2, this was x cube minus 3x square plus 4 whole upon x power 4 minus 8x square plus 16. So this part of mine now has become, after getting this part factorized, this is limit extending to 2, what will we get?

x minus 2. How will this part break? x square minus x minus 2 whole upon x power 4 minus 8x square plus 16 if I am not wrong. And if you focus upon this whole part, this is limit extending to 2, x minus 2. This will be factorized as x minus 2 into x plus 1 whole upon. Look carefully at this part.

What arguments will be there? Here it is written x square of square. Last term it is written as 4 square. Whenever first or last term if it is perfect square, then understand either it will be a plus b whole square or it will be a minus b whole square. Take a chance and see.

No, we will factorize it from splitting the middle term. This term is also perfect square and last term is also perfect square. So, let me see the term of the middle. It is written as x square square.

It is written as 4 square. So, can you write the term of the middle like this? 2 into 4 into x square.

Definitely. So, this is the form of a square plus b square minus 2 ab. So, very obviously and very smartly.

Let's talk about the upper part. This is x minus 2 whole square into x plus 1. and if we look at the lower part, then this is whole square of x square minus 4 again this is a square minus b square Form can be factorized as a plus b into a minus b. So, respectively this part of limit goes off. Limit extending to 2. Upare x minus 2 ka whole square into x plus 1 whole upon. This is x plus 2 into x minus 2 getting whole square if I am not wrong.

And if you solve this part further. So, this limit of mine goes off. x minus 2 whole square into x plus 1 whole upon this is x plus 2 square square into x minus 2 square common factor cancel and as soon as common factor cancel from top to bottom then this factor gets cancelled due to which this limit becomes 0 by 0 or infinity by infinity but they did just say This common factor is cancelled from top to bottom.

So this is the problematic part due to which you were not able to put the limit in the limit. And after this as x approaches to 2, wherever x is 2, 2 plus 1, this is 3, 2 plus 2, this is 4. So limiting value of this function is going to be, sorry this is square, square it, square of 4. So this is going to be 3. So value of this limit would be 3 by 60. So these are few questions based upon factorization method. Next question par move karte hai.

So respectable let's move ahead on to the third kind of method of solving the limit that is rationalization method. Kya special cheez hai? Whenever you write the terms of the root in the question of limit, that is a clear cut indication, in most of the cases you will have to do the rationalization. Why am I saying this?

Most of the cases because when we will be discussing infinite limit, even after the root, we are conscious whether we have to do the rationalization or not. Sometimes in infinite limit, even without rationalization, our limit gets solved. If we look at the basic limit, then if I am given the limit, the terms of root are used, definitely rationalization method seems to be optional. Let's pick the first question, limit x tending to 0, root of 2 plus x minus root 2 whole upon x.

As per my habit, as soon as I get a question of any limit, so respectable, I will take the value of x as 0 or whatever limit is there. And I will directly substitute in this question and see what form I am getting. For example, you will put 0 in x.

Root 2 minus root 2 is 0. If you put 0 below, it is 0. So overall this is going to be 0 by 0 form. And that gives me the hint. You cannot substitute limit right now in the question. Then what should be the plan? Root is visible.

We have to solve the limit. Let's begin the solution. Rationalization seems to be the best option.

root over 2 plus x minus root 2 whole upon x. I multiplied numerator by its opposite sign, which means root of 2 plus x plus root 2. And for making it balanced, I divided by this number, root of 2 plus x plus root 2. And freely we have done so. Let's see what result we are going to receive. This is a minus b into a plus b. Limit extending to 0. Do not forget it in writing.

This is limit extending to 0. a minus b into a plus b. This is a square minus. This is b ka square whole upon.

This is x times what? Root over 2 plus x plus root over 2. Square se root kata. Square se root kata.

So this is going to be. Limit x tending to 0, 2 plus x minus 2 whole upon x into root over 2 plus x plus root 2. This 2 and this 2 are cut, so we are left with x upon this part. And when you cancel x from x, so we are going to have limit x tending to 0. This is 1 upon root over 2 plus x plus root 2. And finally... After doing the rationalization, we will replace x directly with the required limit in the part which I have. So we have replaced x as 0. And as I have replaced x with 0, this is 1 upon 2 plus 0, this is root of x.

2 root 2 this is root 2 plus root 2 is going to be 1 upon 2 root 2 so respectable value of this limit is going to be 1 upon 2 root 2 next question let's see this is going to be limit x tending to 0 x upon root over a plus x minus root over a minus x as soon as any question of limit will respectable first of all I will directly substitute x is equal to given limit that is that is x is equal to zero put zero above also if you put zero below then this is going to be zero so question is giving to me as 0 by 0 means in determinate form and if I get in determinate form by putting limit in question that is a clear cut indication you cannot permissible to substitute the limit what option we are left with I can see the terms of root Means rationalization seems to be the best option Let's see, I just rationalized the numerator Now I will rationalize the denominator But if really you will do so Let's see what result we are going to receive So what will be the rationalizing factor of this? Change the middle sign This is root over a plus x plus root over a minus x whole upon This is x into Root over a plus x minus root over a minus x Getting multiplied with root over a plus x plus root over a minus x Now this is limit extending to 0 x into what? Root over a plus x plus root over a minus x whole upon This becomes a minus b into a plus b what? x times, sorry this is not, this is not x, look at this, this is not x, only above x, below x is not there, take care. So this is going to be a minus b into a plus b, this is a square.

minus b square. This and this root, this and this root will get cancelled out. This is limit x tending to 0. x times what?

Root over a plus x plus root over a minus x whole upon a plus x minus a plus x. a and a will get cancelled out. x plus x is going to be 2x.

So this is going to be limit x tending to 0 x times root over a plus x plus root over a minus x whole upon 2x. x and x got cancelled. And finally, the remaining part, if I substitute x with 0 in that part, so as you substitute 0 as x tending to 0, so this root a plus root a, what will you get?

2 root a whole upon 2. cancellation of this will give you root a so limiting value of this limit is going to be root a in the same way third question i can leave for you if you put limit x is equal to 0 see this function this is a minus a 0 below also 0 do reslization after reslization cancel common factor in this way you can easily get the value of this limit So, respectable is a very small thing to see but definitely this whole process, this whole phenomenon is going to take a very important role while solving the question of limits. So, today we saw what does it mean by say that limit, what is direct substitution, when to remove RHL and LHL, what is factorization method and exactly what is rationalization method. Let's move to the next question. So respectable here it falls.

Some more questions, few more questions. Very nice question based upon factorization method. This is the first question. Limit x tending to 4. x square minus 16 upon root over x square plus 9 minus 5. And this is second question. Limit x tending to a.

Root over a plus 2x minus under root 3x whole upon root over. 3 plus x minus 2 root x. Let's first see how to solve this question.

Let's begin the solution for this question. As soon as I get the question of limit, first of all, I will put x is equal to 4 directly in the limit and I will check it out. Do I get 0 by 0 form? Remember, it is a small habit to see, but in every question, put a limit once and check whether it is 0 by 0 form or not.

I put 16 minus 16, 0. In denominator, I checked 4, 16 plus 9, 25. 5 minus 5, 0. All in all, this form is going to be 0 by 0 form. Indeterminate form. Means, you cannot use limit in this function. Then what should be the plan?

I have seen carefully. I can see a term of root in this. Definitely, I have a hint that I should rest this question. And if really we have done so, let's see.

What result we are going to receive? How did I rationalize? I multiplied the part of the lower part by the factor opposite to the upper part.

So this is x square plus 9 plus 5 whole upon this is root over x square plus 9 minus 5 multiplying with x square plus 9 plus 5 If I look at the bottom, this is a minus b into a plus b going to be this is limit x tending to 4 do not forget it limit x tending to 4 this is x square minus 16 into root over x square plus 9 plus 5 whole upon this is a square minus b square means root over x square plus 9 whole square minus 5 whole square And when you solve it further, limit extending to 4, this is x square minus 16 into root of x square plus 9 plus 5 whole upon this root and this will cancel out. This is x square plus 9 minus 25. 9 minus 25 is minus 16. So this is limit extending to 4. x square minus 16 into this part. root over x square plus 9 plus 5 and this whole portion is getting divided by x square minus 16 this and this common factor got cancelled or just say you can meet us a common factor cancel we are permissible to put the limit directly in the question and if really we have done so if x tending to 4 just see 4 telling you 4 square plus 9 you 25, 16 plus 9, 25's root 5, 5 plus 5, 10. So value of this limit is going to be 10. In this part, I put 4 to x.

So this 4 square, 16 plus 9, 25, 25's root 5 plus 5, 10. So value of this limit is going to be 10. What should be the plan for this? Just like in the question, I saw the limit, so I will first substitute the limit You can see by adding the limit will be 0 by 0 form Definitely we cannot substitute the limit directly What to do? Because I see the root in the numerator and the root in the denominator So I will have to rationalize both Numerator as well as denominator And if really we have done so Let's see what result we are going to receive First let's talk about the numerator, this is like this root over 3x so what will be the rationalizing factor of this? root over a plus 2x plus root 3x to balance this I have to divide it also, let's do it now before that what is the part of denominator?

this is root over 3a plus x minus 2 under root x so what do I need to do the rationalization of this? this is root over 3a plus x plus 2 under root x. To balance this, we have to multiply it above. Let's do it.

Now this is going to be, what will be its balancing factor above? root over 3a plus x plus 2 under root x. And its balancing factor, which I want below, what will be it? This is root over a plus 2x plus root over 3x. Further, you work irrespective.

This is a minus b into a plus b ka form. This is going to be a square minus b square. a square minus b square.

This b square into, the remaining part, what is remaining? This one. These two will be adjusted. a minus b into a plus b. Into what is remaining?

This is 3a plus x plus 2 under root x. And if we come down and see, then this is a minus b into a plus b. What will happen? Again this is a square minus b square. This is 4x, b square 4x into this part.

These two are adjusted here. So this is root over this a plus 2x plus root over 3x. Now 2x minus 3x is going to be this is a so limit x is tending to a 2x minus 3x is a minus x. See below 3x plus x minus 4x x minus 4x minus 3x 3x minus 3x from which you will get 3 out of it.

But you got a minus 6. into, there is a part left above, what is that? This is root over 3a plus x plus 2 under root x. And there is a part left below, this is root over a plus 2x plus under root of 3x.

Common factor is cancelled and as soon as common factor is cancelled from above and below, we are permissible to put down the limit directly. Wherever there is x, we put a there. And write when x tending to a, what will you get?

3a plus a, 4a. This is 2 under root a. Plus this will also be 2 under root a. Come down.

x squared a plus 2a. This is 3a under root 3a. Plus this is also under root 3a. Up here I will get this is 4 under root a. And I will get 2 times under root 3a.

Root will be cut from 2. 4 will be cut. So this is going to be 2 upon under root 3. So 2 upon. And see carefully, 3 is also going together.

Don't forget this. This 1 by 3 is also going together. So this is also going to be multiplied with 1 by 3. So cut from 2. 2 times. So this 3 is remaining below.

This is 2 upon 3 root 3. So this is going to be 2 upon. 3 root 3 so value of this limit is going to be 2 upon 3 root 3 so what is the special thing if you want to see the root of both numerator and denominator without fear apply the process of factorization for both when common factor is cut immediately put the limit and this way you can evaluate the limit based upon factorization method so we have seen today Direct substitution method, factorization method and rationalization method. In the next class we will be discussing about the question, question based upon some standard limit and infinite limit.

Transcript for:Understanding Limits in Calculus

Transcript for:
Understanding Limits in Calculus