 Hi, I'm Zor. Welcome to Unisor Education. We continue talking about function limits, and this particular lecture is dedicated to one side of the limit. I will explain what it is. This lecture is part of the course of Advanced Mathematics, which is presented on Unisor.com and is aimed as a very nice course for teenagers who are interested in development of their logic, creativity, brain power using the mathematics. Alright, so, we are talking about function limits, but not just regular limits, which we have already covered, but the so-called one-sided limits. Now, first of all, what this one-sided limit is from very simple perspective, and then we will go into the details. Well, for instance, you have a function. Now, this is a point which we are interested in. Let's call it x is equal to a, and at that point, we would like to analyze the behavior of this function as we are moving towards this point. Basically, that's what the behavior of the function is a limit when argument goes to certain values. But in this particular case, we are not just going towards that point with any kind of a sequence of argument values. Instead, we are actually prescribing let's move only from the left. So, if our sequence of values is a, and at the same time, it's always less than a for any member. So, we are always going from the left. That's what we are talking about the left-sided limit. And obviously, if we are going from the right when this is an inequality of this type, so when all of these sequences, sequence values are on the right from the point we are talking about, then we are examining the limit of the function as we move in that direction. Now, it's not necessarily to move monotonically. So, it's not like this. It can be as long as we are actually getting closer and closer to the point x is equal to a, which means that for any degree of closeness, there is some number and after which it will be closer than that degree. So, let's concentrate right now on just a little bit more formal definition of what is a one-sided limit. Now, as you know, we were approaching the limits from like two different directions to different types of definitions we have used to define the limit. One is using some kind of sequences of values of arguments. So, right now the definition of that type would be if for any sequence xn, which is approaching a from the left. Now, this is very important for any kind of a sequence which is approaching a being less than a. We have that limit of my function f of xn. Let me just not use the word limit. I'll just use converge. Converges to some number l. If this is true for any sequence which is approaching a being less than a, the function values of the corresponding sequences of arguments is converging to l, then we are saying that and here is a minus sign. You see, it's like a power. Now, that's what it means being from the left. So, it's always negative relative to, not negative if it's to the left of the a. Now, then that's what it means. So, we are defining this as the property of the function f such that for any sequence of arguments which is approaching a from the left, the corresponding sequence of the function values is converging to the l. Then we can say that l is a limit from the left or a left limit, whatever you want, of function f as x goes to a. So, that's the limit from the left. Now, obviously, we can completely symmetrically decide what is the limit from the right. So, that's obviously if for any sequence of arguments approaching a being greater than a, the corresponding sequence of values of the function is approaching l, then we say this is a plus. Then the right limit of the function f as f goes to a is equal to 0. So, these are two definitions of the first type, definitions of the limit of the left or the limit of the right. Now, we also know that we can define the limits using so-called epsilon-delta language. So, remember the regular definition of the limit using this particular language. Let me just remind you. So, if for any degree of closeness epsilon there is such a delta. Now, this is the degree of closeness of the function to its limit. Now, there is a degree of closeness of the argument to the limit value of the argument that as long as x minus a is less than delta, so as long as my argument is closer to my limit value than delta immediately follows that value of the function would be less or less or equal on the distance less than epsilon from the limit value. That's what it means limit f of x, x goes to a is equal to l. So, the definition of this using epsilon delta language is this. For any degree of closeness of the function value to its limit, there is such a closeness of the argument to its limit value that from this follows this. So, as long as my argument is closer to a than the distance delta, my function would be closer to the l in the distance not greater than epsilon for any epsilon, however small. Now, we are talking about definition of this. What is this particular definition? Well, basically it's exactly the same thing, but instead of this we should really make sure that we are approaching the value of argument to the left of the a. How can we say it? Well, that's very easy to say it this way. x belongs to from a minus delta to a not including the boundary. So, this is the open interval so from a to the left. So, if x is within this particular interval from the a, so no more than delta to the left. Then we are requiring that for any epsilon the research of delta as long as x would be within delta on the left from the a, then the function would be less than epsilon from the value l. That's what it is. And if you would like to define the limit on the right, or the right limit, I should change this not from a minus delta to a but from a to a plus delta. So, no more than delta to the right. So, again, for any epsilon, however small we should find some kind of a delta neighborhood of the a. So, if you go closer than this delta but on the right side from a, my function would be less than by epsilon of the limit value l. So, that's what limit on the right. So, these are two definitions which are using the epsilon delta language, which is usually preferable form quite frankly. However, both forms using any sequence and using this epsilon delta both are quite applicable. Now, I would like to prove a couple of theorems which are extremely easy and I do suggest actually maybe instead of listening me right now to pause the video and just try to prove them yourself. The first is the following if the limit of f of x as s goes to a is equal to l then limit of f of x goes to minus l. So, if there is a real limit of the function at point of argument a and its limit is equal to l, then the left limit also is equal to l. Well, this is obviously very, very easy and for this you can use my first type of definition with sequences. Now, any sequence of x and which is converging to a results in f of x and converging to f. That's what this means. Well, if it's any it includes all those sequences which are on the left from the a, right? And that actually proves that the limit on the left is exactly the same. So, if a true limit is equal to l, then the left limit is as well. And obviously the right limit also is exactly the same. So, this is very easy theorem and that's why I suggest you to pause the video before you listen to me and try to do it yourself. Okay, so that's easy. Now, so if function has a limit then it has a left limit and a right limit and they're both equal to the same limit. Now, how about converts? What if the function has left a limit and it has a right limit? And that's all which is, we know about this function. Does it mean that the function has the general limit? Well, the answer is no and a very easy counter example is what if we have a function which is, let's say, equals to 0 before 0.0 and then it equals to, let's say 1. So, at 0.0 it's equal to 0 and to the left and starting from all positive x it's equal to 1. So, what is this, what is the limit of this function from the left? Well, obviously 0 and what's the limit from the right? Obviously 1 and the function has absolutely no limit at 0.0 in a general sense. So, we have left limit, we have right limit but they are different and that's why there is no connection here and the function does not have any limit. It's not continuous in this particular point x is equal to 0. Okay, but what if these two values are the same? So, what if limit is exactly the same on both sides? Well, then we can prove actually that the function actually has a limit in this particular case. And this can be proven in the game, try to do it yourself first, just pause the video and do it yourself. So, right now we are planning to do the following. If you have this and this, so limit from the right is equal to limit from the left and equal to L. Then, plane limit, general limit also exists and it's equal to L. Okay, here is a very easy proof. Now, in this case, I will use epsilon delta language. Now, what does this mean? That this limit is equal to L. Well, it means that for any epsilon greater than 0, there is such a delta, which I will use in index 1, such that as long as my x belongs to, this is plus, right? So, it's from a to a plus delta. Then the function would be within epsilon from L. Now, what this means? Well, this means that for any epsilon greater than 0, there is such a delta 2 that as long as x belongs to, it's on the left, it's minus, right? So, it's from a minus delta to a. We have this closeness of the function to its limit. Okay, so, that's what we know. That's given, basically. Now, what do we have to prove? Well, we have to prove this which means what? It means that for any epsilon greater than 0, there is such a delta that if x minus a less than delta, then function minus L, but it is less than epsilon. That's what we have to prove. Well, let me just write this slightly different. It's x from a minus delta to a plus delta, right? That's what it means. Now, how can I prove that? Well, let's pick any epsilon, find delta 1 that this is true, find delta 2 that this is true. And now, let's take delta is equal to minimum of delta 1 and delta 2. Now, if it's less than delta 1 and delta 2, it means that both are true statements, right? And that's exactly what this is, because if you combine this statement and this statement, so it looks like if my x belongs to delta neighborhood of a, it means that this is true. Oh, I forgot to put 1 and 2 here. I'm sorry. So, it means that this is true and this is true, which means this is true and this is true. So, for any x which belongs to this interval, we have this thing true. All we have to do is to pick up delta 1 from the right and delta 2 on the left and have a delta neighborhood which is narrower than delta 1 on the right and delta 2 on the left, right? This is delta 2 or delta 1 on the right is delta 1, right? Delta 2 is on the left. Now, I take the minimum which is, let's say, this one. So, this is my delta neighborhood. Well, the only thing which I think needs just a discussion. Now, when I'm saying that x belongs to this delta 1 neighborhood, does it include this point a or not? Now, the same thing actually is what do we really understand as a limit whenever some sequence of arguments is approaching a. Does it mean that it's allowed to take the value a? Now, this is something which you just can pretty much pick yourself. It depends on definition as we say. So, if my sequence f x n approaches to a, does it mean that the value a is allowed to be taken by this sequence or not? Because if not, then the function like this. For instance, the function is like this and it continues like this, but at point a it's something else. So, everything less than a is this. Everything greater is this. It doesn't matter what kind of curve this is. Now, they are actually going to the same point. But at value as x is equal to a, the function is going to take some other value. Now, is this a function? Yes, it is a function. If we are talking about limit on the left which does not include this point, point x is equal to a then the limit would be this value. If you go from the right, same thing. If you go in any sequence, if you are not allowing x is equal to a value of argument, then no matter how you approach this a, you will always have the same limit. And the function basically has a limit equal to this particular value. But the value of the function at that particular point is not coinciding with the limit. So, it all depends on how you view this type of limit problem. Usually, we are considering this not to be a function which we will be researching or analyzing or anything like that. You can just think that these functions do exist, but usually we mean that the function is actually continuous which means that the value of the function at this point is exactly the same as the limit as argument is approaching that point. But again, that's kind of a decision which we are voluntarily, usually make. So, in the future, most likely functions like these with one particular point somewhere else not coinciding with the limits from both sides, we usually do not consider these functions. However, again, you should know that they exist and to define it is very easy. You can define something like function f of x is equal to sin x for x not equal to, let's say, 0 and it's equal to, let's say, 1 for x is equal to 0. Is it a function? Yes, it is a function. Now, limit of this function as x approaching 0 from left, from right or whatever else you want would be 0, right? But the function sin itself is equal to 0 and function sin is continuous. But if we are defining our function as coinciding with sin everywhere outside of 0 and at 0 it's something else. And it's a different function, it's not a sin anymore, obviously and it has this peculiar type of behavior at one particular point. And again, we usually will not consider them. So, basically, all I wanted to say that there are one sided limit, left side limit, right side limit and two important theorems that if the function has a regular limit, general limit, then left and right limits exist and not exactly the converse theorem, because if limits exist, left and right doesn't mean that the function have general limit. But if they exist and coincide both left and right, then actually the function itself has exactly the same limit. That's it. I suggest you to go to Unisort.com and read the notes for this particular lecture. They're very useful. Other than that that's it. Thanks very much and good luck.