 Hi, I'm Zor. Welcome to IndieZor Education. I would like to start getting closer and closer to derivatives, and the next stop is talking about function limits. Actually, we have already considered this particular topic function limits when we were talking about the limit of the sequences. I deliberately included a few lectures about sequence limits to this particular part of the course, which is basically dedicated to derivatives. And the whole course, by the way, is presented on unizor.com. I do suggest you to view this course from this website, because, number one, it contains very detailed notes for each lecture. Number two, so it's basically like a textbook. And number two, registered students can take exams, which is just very useful for self-checking. Alright, back to limits. So, as I said before, we have already touched the limits when we were considering the limits of the sequence. But let's just think about what E is as sequence. Well, sequence is a function. It's a function which has as a domain all the natural numbers, one, two, three, four, five, and the values are some real numbers. Alright, so we can say that this is a function from the set of all natural numbers to set of all real numbers. Now, there is only one more very important consideration when we are talking about limits. Talking about limits implies talking about certain process. It's not like a single value of something. It's a multitude of values which are going somewhere, right, to infinity or to some natural number or whatever. Now, what is the process in case of a sequence? The process is one, two, three, four, etc. So we are moving within our domain of the natural numbers from any number to the next one, to infinity. So this is the process. So whenever we are talking about more general functions and in particular we will be talking about functions which will be of that kind. Domain is all real numbers and the codomain is also real numbers. That's the only difference between the function and the sequence. But again, there is a second aspect of talking about limits, the process. So in sequences we understand what this process actually is. It's moving towards increasing natural numbers. In this particular case we don't really fix this particular process. We don't imply it. If we are talking about the limit of the function so far we have not implied any kind of a particular process of changing the argument. So that's something which we need to basically introduce. In addition to the fact that this is a function of the real argument and real values we have to introduce the process, how argument is changing. Well, it's changing according to some rules, according to some sequence of values which argument takes. And the simplest way which resembles exactly the situation with sequences is to consider this process of moving our argument, making it infinitely large. So if we imply this particular process to the function of real argument then actually there is no difference between this and the sequence. We can define a limit in exactly the same terms as here. Here is how. Let's just recall again how the limit is defined in this case. Now what does it mean? If you remember we want it to be as close to A as possible whenever my index is large enough. More precisely, for any degree of closeness to this value we can find the number, the index such that all the members of our sequence will be closer than that particular value of closeness which we want. Let's define it in more abstract terms using symbolic. So for any epsilon greater than zero this is the degree of closeness we would like to be to the number A. However small doesn't really matter as long as it's positive. So it can be as small as possible and we don't really care how small it is. For any however small we can always find or there exists such number N. Now this N is just a number, it's not a set of natural numbers. So there is a number N such that as long as our index is greater or equal than N immediately from this would follow that our X N minus limit less than or equal to epsilon. Sometimes people are using strictly greater and strictly less. It doesn't really matter, it's the same definition, the same property. So this is the definition of the sequence for our sequences. We can actually make very very close definition here. Namely for any closeness to some kind of a limit which we are talking about. So we are assuming that we want this to converge to the A as X goes to infinity. That's what we are trying to define basically right now. So for any degree of closeness greater than or equal to 0, I'm sorry. There is always, now instead of number N of a sequence I can always have some value. Let's say A, some value A such that as long as X exceeds this value immediately my function of this X would be close to the A by the distance less than epsilon. This is exactly the same as this one. So instead of using the index, I'm using the value of the argument itself. Well, actually index is an argument in this particular case, right? Because this is the function from natural numbers to real. This is the function for real numbers to real. So this is the real number and instead of index number N I'm using some kind of a real number A. So as long as we exceed that number from that time on my function would be closer to its limit than the value of epsilon. And epsilon can be chosen completely, any value even the smallest one possible, as small as possible. So that's what it means that the function goes to its limit A converges to A when X goes to infinity. Now, absolutely symmetrically, which does not really have an analogous with natural numbers, we can have when the X goes to minus infinity. Well, why? Because, well, natural numbers are starting with one goes to only plus infinity. But here we can go in exactly the same fashion to minus infinity. How can I say, so this is as X goes to plus infinity, right? Now, what if I'm talking about minus infinity? Well, this is the movement, right? Goes to minus. So here is minus infinity. Let me write it cleaner. Negative infinity. It means we're going to a smaller and smaller number and there is no limit which limits us as far as argument is moving to the minus infinity, right? So it's infinitely growing by absolute value, staying negative. So that's what it is. So whenever we have this type of a movement and we are looking for a definition of this, when X goes to minus infinity, what does it mean? Well, the only difference is I have to choose this A on a negative side and all the X which are less than this negative A, this closeness is supposed to be observed. So for any degree of closeness, however small, there should always be number, real number A, which obviously in this particular case should be on a negative side, right? Such that as long as X moving towards minus infinity, as long as X is less than to the left of the number A, negative number A, our difference between the function and its limit would be less than the epsilon. So these are two definitions of the limit of the function as argument goes to either plus or minus infinity. And these are very much parallel to the sequences. Okay, now let's consider that we would like actually to expand it even more and that is completely impossible with sequences. We always have M goes to infinity and that's why we started for function in exactly the same fashion. But in real functional analysis and if we want to talk about derivatives, not only we are interested in limits as the argument goes to plus or minus infinity, but also as the argument is approaching certain concrete real value, converges to a value. And I would like actually to go basically the same logical way. So what can we say about, that's what we would like to define as X to some number R, not to infinity plus or minus, but to some number. So X is converging to R. Well, what does it mean converging? Well, for instance, X actually has a sequence X1, X2, etc., Xn, etc., and this sequence is converging to R. Now, if that is the case, then we would expect that the corresponding f at X1, f at X2, etc., f at Xn, etc., is converging to A. Now, we have actually transformed the definition for the function into a definition for sequences. So what I'm saying is that if a sequence of arguments goes to R, then my sequence of functions of these is supposed to converge to A. Now, this is something which I would like to use as a definition, but I can't. And here is how, and here is why. I mean, I cannot just leave it as this. I'm saying, okay, if my arguments are converging to R and my functions of these arguments are converging to A, then I can say that the limit of the function at point R is equal to A. This is wrong. I have no right to do this. And here is why. Let me just explain to you, for instance, how this is possible. Well, let's just have a concrete example. Xn, Xn is equal to, let's say, R plus 1n. For instance, as n goes to infinity, Xn goes to R, right? And my functions are, and I expect them to be converging to A, right? As n goes to infinity. Now, on the other hand, I can have another function, another sequence which approaches R. Let's say R plus square root of 2 divided by n. As n goes to infinity, this also converges to R, right? Because square root of 2 over n is infinitesimal. Now, my functions in this case would be, now, square root of 2. Now, am I guaranteed that this is true? I mean, if this is true, am I guaranteed that this also would be true? Or any other way how X can approach R? Will we always be having this type of a converging to the same number on the app side? Well, the answer is no. Here is example. Consider the function 0 if X is rational and 1 if X is irrational. And now consider X going to 0 by equal to 1 over n. Now, 1 over n goes to 0, that's true. I mean, I should say Xn, right? So, Xn goes to 0 using rational numbers, right? 1 first, 1 second, 1 third, 1 fourth, etc., etc., converges to 0. And my function would be equal to, on each rational number, it's equal to 0, right? So, my function is equal to 0, so it converges to 0. Now, let's just have another example. Square root of 2 over n. Now, this is irrational number, right? So, every value is equal to 1, and it's converging to 1. So, it looks like the limit depends on how we approach 0. If we approach 0 on rational points, it's one limit on irrational and another limit. That's not good. We cannot define, therefore, we cannot define that the function has certain limit if approaching the limit point by one way we have one limit and by another way another limit. It just basically means that the function does not have a limit. So, how can we correct this particular definition to make it really valid? Okay, here is the valid. f at x goes to a if r converges to, sorry, if x converges to r, if for any, here is very important, for any sequence converging to r, the corresponding sequence of the functions converges to a. So, the key here is this. So, not only one concrete sequence should go to r, and then we see the sequence of the functions converging to a. But if for any possible sequence of arguments which converges to the limit point, to this value, to the limit point, if for any such sequence of arguments, the corresponding sequence of functions is converging to real number a, then we can say that function converges to a when x converges to r. So, that's very, very important. We should not tell this if we observe this particular behavior for one particular sequence of arguments. We really should say that if for any sequence of arguments this is true. As long as the sequence of arguments converges to r, this converges to a for any sequence. Only then we can say that the function has a limit a at point r. Now, what's good and what's bad about this? Well, the good thing is that it's a really very good definition. The bad thing is, okay, how can I check that particular function has a particular limit at a particular point? Well, I have to basically go through all the possible sequences of the argument which lead to this point and check that for each such sequence, I have the function sequence which is converging to a. Well, that's impossible. I cannot. Which means that I need some more constructive definition of the limit which resembles the one which I did for infinity. You see, for infinity all I had to do for any degree of closeness I had to find the boundary after which my function would be closer to its limit than this particular degree of closeness. Actually, I can use exactly the same approach in this particular case which would lead me to an alternative definition of the limit. And let's try to do it this way. So, we are looking for different definition but would basically mean the same thing but in a more constructive way. So, what do I want to do? I want to approach my argument towards the limit point. So, I want x approach r and I would like as a consequence of this to have function of x approaching its limit a. Now, so how can I express this? Well, as any sequence, I just did, any sequence of arguments which is approaching r. But I can do it differently. I can do it the same way as I did for infinite limits. First, we choose any degree of closeness between this and this. Now, instead of saying that for any sequence which is approaching, any sequence of arguments which approaches r, I can say different as long as x is closer to r. So, if for this particular degree of closeness, instead of finding like in the sequence number n by which my sequence would be closer to its limit, I would say I will find degree of closeness of the argument such that if my argument is closer than this, then my function would be closer than that. So, let me just repeat it again. So, there is analogy with definition for cases when x goes to infinity. In this case, I'm basically measuring, instead of how close to infinity my x is by putting the boundary after which it's closer than we need. I'm putting this closeness using this particular inequality. So, as long as my x is closer to the limit point than this particular delta, then my function would be closer to its limit less than the degree of closeness which I'm looking for. And that should be predicated with this. That's very important. So, for any degree of closeness of the function, there is certain neighborhood, if you wish, of this limit point r that if I am within this neighborhood, if the difference between them is less than this delta, then function would be in the neighborhood of my limit. So, this is alternative definition of the limit. It basically means exactly the same thing. The equivalence of these two definitions, one through any sequence which is converging to r, another through this neighborhood concept. The equivalence of these two definitions will be addressed in the next lecture. But in this lecture, I would like actually to stop exactly here. I would like to finish with these two definitions of the limit of the function which are equivalent. Now, I consider this to be a little bit more constructive definition. Why? Because given the epsilon, given the degree of closeness of my function to its limit, I actually can find, if I can find such delta that from this follows that, it means that the function has its limit. Now, let me just give you an example. For instance, I am looking for limit of function x square as x approaching, let's say 1. Now, we feel that it should be 1, right? Because 1 square is 1. But let's just think about it. My x is approaching 1 and I am looking for, given my epsilon, I am looking for such delta that if x minus 1 less than delta x square minus 1 would be less than epsilon. Can I find this? Can I find, for each epsilon, can I find delta? Well, let's just think about it. If x minus 1 is less than delta, it means that x is in these boundaries, right? From 1 minus delta to 1 plus delta. Now, x square would be from 1 minus 2 delta plus delta square and 1 plus 2 delta plus delta square. I would like this to be true, which basically means that x square should be 1 minus epsilon and 1 plus epsilon. So, how can I do that? If I know that my x is in these boundaries and I know that x square, my x is in this, x square is in this boundary, basically I have to really do this. It is epsilon equal to delta square plus 2 delta on the bigger side, right? And on this side, is this sufficient or not? Let me just check. Let me just try on this side first. So, if my delta can be found from this particular equation, so for any epsilon, I will find delta which satisfies this particular equation. Then I can say that my x square, if x is less than 1 plus delta, then x square will be less than 1 plus epsilon, right? So, that's all I have to do to find delta. Now, on this side, I have to find, let's call it delta 1. On this side, I will have its minus, right? So, minus epsilon is equal to minus 2 delta plus delta square. And this would be my delta second. So, I will solve this equation. And when I solve these two equations, I have two different deltas, I will choose the smaller one and that would be sufficient to satisfy my both equations. So, that's basically how this thing is solved. I'm really finding, I'm saying that the definition of the limit is, for any epsilon which is the closeness of the function, I should find the delta which is restricting my argument as close to the limit point as possible. So, this is how I can find a concrete delta for any epsilon. And that's the advantage of this second methodology, that we can always find exactly this particular delta in cases when the limit exists, obviously, like in this case. And if we can find, for any epsilon, we can find this particular delta, then it means that the function does have the limit 1, in this case, if argument is converging to 1. Alright, that was my introduction to the concept of the limit of the functions. And the limit, basically, we are differentiating between three different points. Two are very analogous to each other when argument goes to infinity or minus infinity. And the second, when the argument is approaching certain concrete number, and in this case we have given two different definitions of the limit. One is like for any sequence of arguments which is approaching the limit point, we have the corresponding sequence of functions approaching to, converging to its limit. The second is using this as they call epsilon delta language. So, an epsilon delta language, which is a different definition, although equivalent, we have expressed it differently, but more constructively in such a way that we can actually find and prove that this particular function has this particular limit under these particular conditions of argument. Alright, I do recommend you to read the game through notes for this lecture on Unisor.com, and basically that's it for today. Thank you very much and good luck.