 Hi, I'm Zor. Welcome to Unizor Education. Today, we will just make a couple of examples of the limits for functions, function limits. This is a very, very simple introduction to problems related to function limits. And these problems are really very, very simple. It's just illustrative examples rather than real problems. Now, this lecture is part of the course of advanced mathematics for teenagers and high school students. It's presented on Unizor.com. I suggest you to watch this lecture from this website because the site has very detailed nodes for each lecture, including this one, and also registered students can take exams. They can enroll into a specific topic or entire subject and then take exam on different parts of this course. Okay, so illustrative examples or simple problems, if you wish, of the function limits. First of all, let me remind you something which I have discussed in the previous lecture, two separate but equivalent definitions of the function limit. The first definition is based on definition of the limit for sequences. So, you remember for sequences, we are saying that a sequence is converging to certain number a if for any, any is very important. For any positive epsilon, we can get closer than epsilon to a after certain order number. So, there is such a, there is, there exists such number n that if my order number is greater or greater or equal to n, doesn't really matter, then from this follows that the distance between them would be less than epsilon or equal. Sometimes I'm using equal, sometimes equal or less, doesn't really matter. Okay. So, that's the sequence. Now about function, now the function definition of the limit can be based on this one, and we are saying that the function, the function f at x converges to some number a if, and this is now very important, for any sequence of arguments converging to some number r as, as x converges to r. So, we are defining that function is converging to a if its argument converges to r, if for any sequence which is converging to r, the corresponding sequence of function values converges to a. And this is very important. So, for instance, this is our function graph, this is our r, this is our a. So, no matter how we approach r from left, from right, from left and right, etc., if approaching r necessitates approaching of the function value to this a, then we are saying that the function has a limit. And that's where, what's very important is this word any. So, any sequence which is converging to r of the arguments should cause the corresponding sequence of functions to converge to a. So, this is one definition of the function limit. Now, another definition not based on sequences, but based on so-called epsilon delta language, which means basically exactly the same thing but expressed differently. So, we are saying that the function f at x converges to a when x converges to r whenever for any epsilon greater than zero, for any, that's the symbol for any, for any closeness, so to speak, of the function to its limit. There is always such closeness of the argument that if x is closer to r than delta, then f at x would be closer than a less than epsilon. So, again, we are not using the definition of the limit for sequences, we are using just plain functional elements, argument and the function. So, for any closeness which we want between the function and its limit, there is a corresponding closeness of the argument to its limit, such that if my argument is closer than this, then function would be closer than this. Now, and I proved that these two definitions are equivalent. So, we will use these definitions to prove certain examples of the limits. And my first example is that the function f at x equals ax where a not equal to zero has a limit of zero when x converges to zero. So, that's what we are going to prove. The limit of the function f at x equal to ax, where a is some constant not equal to zero. The limit of this is zero at the point of the argument zero. So, how can we prove it? Well, we can actually use both ways, both definitions of the limit. So, let's first concentrate on the definition related to sequences. I have to prove that for any sequence xn converging to zero, the corresponding function sequence also converges to zero. How can I prove it? Well, obviously, this is a times xn. And we know from the properties of the sequence that if my sequence goes to a certain limit, then my constant times this sequence goes to this constant times the limit. The limit is zero, so it will be zero. So, that's how we basically approach this from the definition using the sequences. Now, can we use the second definition, the epsilon delta definition? Yes, absolutely. Let's just do that. It's very simple too. So, I have to prove that this is goes to zero if x goes to zero. So, let's choose any epsilon and we would like to be closer than epsilon between ax and zero. How can I find such delta that from this, sorry, from x minus zero follows this? How can I find such a delta? Well, first of all, minus zero can be completely skipped, right? Here and here. So, from this, we have to do this. Well, very simple. Delta is equal to epsilon divided by absolute value of a, right? So, if my x is less than epsilon divided by absolute value of x, what follows absolute value of a times x less than epsilon, right? And that's exactly what we need. So, from this follows this. So, we have proven basically that ax converges to zero if x converges to zero using both definitions. The sequence base definition of the limit of the function and the epsilon delta language definition. All right. So, basically this is finished. Another example, very, very similar for function x square. Again, we want to prove that the limit of this function is zero if x goes to zero. Again, first, let's try the sequence. Let's choose any sequence xn which goes to zero. How can I prove that corresponding sequence of the functions which is actually x square goes to zero? I have to prove this. So, this is given, this is proved to be proven. Well, I mean intuitively it's obvious, right? xn is infinitesimal than xn square which is a small times small would be even smaller than that, right? On another hand, let's just think about this in terms of like more mathematical approach, obviously. Now, what do we have to prove here? We have to prove that for any epsilon greater than zero exists such n or let's put m. This such m that if my m greater than m, it follows that xn square would be smaller than epsilon. So, how can we find such an m? Well, let's do it this way. I know this which means that for this epsilon, for any epsilon greater than zero, there is such an n that if my n greater than n, my absolute value of x n greater than or of epsilon, right? So, this is given and this I have to prove. All right, so choose any epsilon then using this property and instead of epsilon, I put square root of this epsilon. For this also exists such n that if n greater than n, absolute value of x n and therefore if I square this, I will have x n square greater or less than or equal to epsilon. So, all I have to do is choose any epsilon, take square root of this. It would be another small variable, right? Using this for this epsilon equals to square root of this epsilon, I will find n. So, my xn would less than square root of this epsilon, which is this one and then immediately follows that xn square. So, for this, for any epsilon, we have found the n, so this one would be true. So, this is the sequence definition of the limit. Now, how about epsilon delta language? Well, probably it's as easy. So, let's choose any epsilon. Now, I have to find delta. So, let me just put delta equals to square root of epsilon. Now, for this particular delta, which I found based on this epsilon, if my x is less than delta, which is square root of epsilon, immediately follows that x square would be less than epsilon. So, that's as easy. Again, we are using the definition. So, definition requires that for any epsilon, I have to be able to find such a delta that if this is true, then this is true. Okay, I found the delta. It's square root of epsilon. Very simple. Okay, and the last example, again, in the relative is simple, but a little bit may be different. My last example is log 2 of x. Now, if x goes to 1, then this is supposed to go to 0. Remember the graph? This is 1 and this is log 2 of x. So, if x approaching 1, the value of the function should approach 0. So, I have to prove this. Okay, same thing. Let's just use epsilon delta. Choose any epsilon greater than 0. What do I need? I need that my log 2 x should be, I have to find such delta that if x is within delta neighborhood of 1, then this is true. Right? That's what I have to find. I have to find delta. So, in another notation, I can put minus epsilon log 2 x plus epsilon. That would be easier, right? That's exactly the same thing as this one. You agree, right? Or if I will use exponential function 2 to the power, it would be this. What is 2 to the power of log x by the base 2? That would be x, right? So, this is equivalent. So, I have to find, for my epsilon, I have to find such a neighborhood of 1 that if x is within that neighborhood. Now, what is 2 to the power of minus epsilon? Well, epsilon is positive, right? So, this is just slightly less than 1. It's somewhere here. This is 2 to the power of minus epsilon. And 2 to the power of epsilon is on the right side. So, this is basically, if my x is between this and this, then my log x would be less than epsilon, right? So, how can I choose the delta so that delta neighborhood of 1 would be exactly what I need? Well, how about this? I will take 1 minus 2 to the power minus epsilon. That's this piece. And 2 to the power of epsilon minus 1. It's this piece. And I will take minimum of these 2 and call it delta. So, if this is minimum, it means my 1 minus delta and 1 plus delta would be within this interval. So, 1 minus delta and x 1 plus delta is smaller than this one. And that's why this would be definitely true. And that's why this would be definitely true. So, again, how can I choose delta? This is how. I take the minimum between 1 minus 2 to the power of minus delta, which is this. And 2 to the power of epsilon minus 1, which is this. Minimum of them is delta. And if x is within delta neighborhood of 1, which is from 1 minus to 1 plus delta, then necessitates this also is true. And that's why this is true. And that's why this is true. So, basically, we have proven that for any epsilon, we have found such a delta that as long as x is within neighborhood of 1, delta neighborhood, then function would be in the epsilon neighborhood of 0. Okay. So, these are extremely simple examples. And I wanted to present them just as an exercise of the logic of how to prove that something actually has a limit based on the definition of function limit. I do suggest you to read the notes for this lecture. It's on Unisor.com. And, well, basically, if you will feel comfortable about this, then, by all means, follow the next lecture. Thank you very much and good luck.