 Hi, I'm Zor. Welcome to UNISOR education. I would like to start talking about things related to calculus, primarily this chapter is that is dedicated to derivatives. So this is the first lecture of this chapter of the course. The course is presented on UNISOR.com. That's advanced mathematics for teenagers and high school students. I do recommend you to view this from this website rather than from like YouTube or any other source because the website contains detailed notes for each lecture like this one and also for registered students. It contains exams which they can take, they can enroll into different courses under supervision of their parents or teachers or whoever. All right, so back to derivatives and I was just thinking about what is the proper way to introduce derivatives and the first thing which probably I would like to pay a lot of attention is everything about limits. Now we did address the concept of limits for sequences in algebra part of this course. In this lecture and a couple of others which will follow I will well, generally I will repeat a little bit, maybe briefly, whatever I was talking about in those lectures. But I do suggest you actually to go back to these lectures in the algebra. It's called limits basically where I present certain proofs of very simple theorems, which I will just mention today, but I'm not going to prove them. For the proof I would probably refer you back to these lectures in the algebra course. And primarily it was in the algebra for the purposes of arithmetic and geometric progression. Here we are talking about limits in general. It will be a little bit more rigorous, but in any way it's just a continuation. So first we'll talk about limits of sequences. So number one, what is a sequence? Let's try to be as rigorous as possible in our definitions and formulations, statements, etc. So what is a sequence? Well, sequence is a set of elements, and I put the number n. It means that this set is countable, it's ordered, and means that there is an element number one, element number two, element number three, etc. So countable, ordered, ordered, and it's also infinite, what's very important, and actually goes to infinity, as we are saying. We'll talk about what does it mean to go to infinity, but I think intuitively everybody understands that for every natural number from one to two without any limits. There is an element of this particular set. Okay, so that's the sequence. Now, what does it mean that the sequence has a limit? Or we are talking in other way sequence convergent to some limit. Alright, so we are talking about convergence, the sequence has a limit, let's call it L, and this is the definition. Now, let me just start with intuitive definition. Well, it means that the values of this sequence, as n is increasing, are getting closer and closer to L. Now, this closeness should be really going down to zero, and here is the way. Consider, for instance, this is L, and these are a one, a two, etc., a n, and they are closing and closing and closing, but they are actually do not step over number m. Does it mean that they are going to L? No, they are getting closer, but not that close as we would like it to be. We would like it to be as close as possible. Well, how can we formulate it rigorously? Well, let's just take any kind of a positive constant epsilon. However small, that's what's very important. So it's any constant, but we are not actually restricting the range. We are saying it can be any positive constant, however small, and then we should say that eventually these a nth should be closer than epsilon to L, right? Which means that absolute value of the difference is less than or equal to epsilon. So that's our purpose. How can we formulate it? Well, number one, we should really worry about another case. What if our sequence goes to L and then goes back? Is it possible? Then it goes L to L again, very, very close, closer than the first time, and then out. Well, think about the sine, for instance. The graph of sine, it goes to zero, but it doesn't mean that it has a limit of zero, because it goes further. So as our values are going forward, we are getting closer to something, and then we are getting out of this closer again and getting out. So can we say that the limit of sine is equal to zero? Generally speaking, not. It all depends. I mean, if we are limited to this particular interval, then yes. But if we are not limited to this interval, then no. So basically, we would like our sequence not only to get as close to L to its limit as possible, but we would like it to stay in that neighborhood of L, which is defined by the value epsilon. So epsilon defines a neighborhood around L. The smaller epsilon is, the narrower this neighborhood is. So we would like, eventually, for our sequence, not only to fall within this L minus epsilon and L plus epsilon within this neighborhood of L, but we would like it to stay there where from there on and without getting out. Combined with basically a freedom of choosing epsilon and epsilon can be however small, it means that even for the smallest interval possible, we will still be eventually within this neighborhood of L and we will stay that way. That's what basically the definition. Now, how can we formulate it better? Here is the way. For any epsilon greater than zero, we should find such number n, so exist such number n, natural number n, such as L minus a n less than an epsilon if n greater or equal to n. So starting from certain number n, our sequence would be closer than any epsilon as soon as our number n is greater than this n. Obviously, the smaller the epsilon is, well, most likely the larger n should be because it will be later on when we will get closer than epsilon to L. And if we are reducing epsilon, that's more strict requirement. So it might actually be a little later. But eventually we will come. So the sequence has limit L or we are saying convergent to its limit L. If this condition is true, which means for any positive epsilon which defines the neighborhood of this limit L, exist such a number n after which after which our members of the sequence will be closer to its limit than this epsilon. So that's the definition. It's quite rigorous, so you can actually use it and we will use it in the future for limits of the functions. All right, now examples. Well, obvious example is one n. Now, what happens as n increasing? As n is increasing, we will have one, two, one, three, one, four, etc. One over a million, etc. And what is the limit of this? Well, obviously the limit is zero. It gets closer and closer to zero. And I can actually count for each epsilon, however small, I can count actually when my sequence will be closer to zero than epsilon. Now, so my one over n minus well, we usually do it L minus. So it's L is zero minus one n by absolute value should be less than or equal to epsilon. Now, the absolute value of this is one n. So it should be less than epsilon, which means n should be greater than one over epsilon. So whenever my number n is greater than one over epsilon, this would be a true statement for any epsilon. So if the epsilon is very very small, this will be very very large, which means it's much much later when we will be closer to this smaller epsilon, but it will be eventually and it will stay that way because if n is greater than one epsilon, then always for any n even greater than this one for any n my one over n would be less than epsilon. So that's a sample that's kind of an example of a case when the sequence has an exact limit and we basically know and we have proven what actually the limit is. Are there any sequences which do not have any limits? Well, obviously. Consider such a simple thing as one minus one, one minus one, etc. So it's one minus one to the power of n plus one. That's my sequence. So if n is equal to one, I have n plus one two, so minus one square would be one. If n is equal to two, I will have cube, so it will be minus one. Then the fourth degree, same thing, one. So now does this particular sequence has a limit? Well, no. It jumps over from here to here, from here to here. It does not get permanently close to any number. So it's very, very often when it's equal to minus one and it's very, very often when it's equal to one. But it's changing, which means it does not have this particular number n. After which it's close to either one or another one. So it's always changing. So this is sequence, which does not have a limit. And obviously, there are other sequences which do not have a limit. Now, next. Next, I promised to talk about certain properties of the limits, which I have proven back in the lecture on limits in algebra course. So I'll just list them without any proof. And they are very, very obvious and very, very easy. Okay. Theorem number one. If a sequence is convergent to some limit, then it's bounded. What does it mean it's bounded? Well, it means that absolute value of n would be less than some value a after certain number n and following. Why? Well, we can choose a to be greater than absolute value of l, right? Now, our values are concentrating around l and very, very close to l. If we will take a large number n and all subsequent numbers, right? So if we will skip all these first numbers before it falls into some into some neighborhood of l, which is finite, which means a can always be above any of those initial values of the sequence. So after that, it will be within neighborhood of l. So this is l and this is a. So eventually all our values would be concentrating here. We can choose this neighborhood close enough so it does not go up to the a, right? So basically accept certain number, finite number of initial values, which are here. All other numbers will be here, which means that there is a boundary. Now, this is obviously a boundary on this side. And since these guys are finite, there is always minimum and maximum. So we can always have minimum and maximum. And it's always bounded. So it does not go to any kind of an infinite value. OK. By the way, maybe a little bit more complicated example. Of the segments, which does not have a limit, but it goes to not to minus 1 and 1. This is kind of a primitive thing. Here is what I suggest. 1, 1 over 1, 2, 1 over 2, 3, 1 over 3, 4, 1 over 4. So you see these are going to 0. And these members are infinitely increasing. So it's sequence, this particular sequence goes to 0 in a large number. Again, closer to 0, even larger number. Even closer to 0 and even larger number. So it's also non-convergent, but in a little bit more complicated way than my first example where it was minus 1 and 1. So examples of convergence can be or non-convergence can be very sophisticated. All right. So my first theorem was that the convergent sequence is bounded on both sides. Now next is very simple one. If a particular sequence has a limit, then sequence of the same sequence, but every member multiplied by some number is also convergent. And the limit is the previous limit multiplied by the same number, the same factor. Next, next if I have two sequences and both are convergent to different limits, then some of them would be convergent to some of limits. Or we can say limit of sum is equal to sum of limits, so to speak, right? Limit of sum is equal to sum of limit. Now, next is absolutely similar, but with a product. Again, for the proof of each one of them, I can refer you to lectures in algebra where I investigate the limits for the purpose of sequences. And actually it would be much more beneficial for you if you try to prove these theorems yourself first. So what does it mean to prove this particular theorem? Well, for this particular case, for instance, well, you have to choose epsilon and then you have to find number n after which this minus this by absolute value would be less than epsilon. So you have to find that for every epsilon, you will find this number n. So that's your purpose in your proof. Next, next is if this is true and l not equal to 0, then 1 over a n converges to 1 over l. Also kind of intuitively obvious thing, but you obviously understand why I chose not 0 because if I do, then this would not exist, right? And finally, if you have two sequences and you know that m is not equal to 0, then a n over b n converges to l over m. And obviously this is for the same purpose because it's denominator, right? So these are sequences. These are theorems about these sequences, about the limits of these sequences. And the only thing which is remaining, well, let me just give a couple of examples. Just as an example, if I have something like this as a sequence. Now, my statement is that it goes to 1. How can I prove it? Well, let's just think about it. That's what this is, right? n plus 1 divided by n is equal to 1 plus 1 nth, right? If you will take it to common denominator, this would be n over n, 1 over n, so n plus 1 over n. Now this is the constant. So with this constant, I'm associating another sequence. So basically this is sum of two sequences. 1 is always 1 as a constant. It's also a sequence, right? 1, 1, 1, 1. Does it have a limit? Yes, the limit is 1. And the difference between 1 and 1 is 0 always. So no matter how small you choose epsilon, all members would be exactly where the limit is. So obviously the distance would be less than epsilon because it's 0. Now 1 over n, as we have already discussed before, it goes to 0, right? So this goes to 0. This goes to 1. And by the limit of the sum of two sequences, the limit is supposed to be 1 plus 0, which is 1. Okay? That's simple. Now a little bit more difficult example, something like this. Two polynomials, 2n squared plus, doesn't really matter. 5n minus 1 divided by 5n squared minus 7n plus 2, whatever. Doesn't really matter. I don't mean to have any kind of common multipliers or whatever. What is the limit of this sequence? Well, I will be talking about polynomial and ratio of two polynomials separately. But in this particular case, let me just state that I can always convert 12n squared plus 5n minus 1 as 12 over 5 times this one, 5n squared minus 7n plus 13. Plus some multiplier and some multiplier, some free member. Why? Because you see, 12 over 5 and this 5n squared. So if I will multiply, it will be 12n squared. So I have this one. Now all these guys are n to the first degree and the constant. So I can always find a to take 12 over 5 minus 7 to be equal to 5 and this to be equal to b. So it looks like this thing is equal to 12 over 5, 5n squared minus 7n plus 13, right? Divided by 5n squared minus 7n plus 13 plus another sequence minus 7n plus 13. Now in this case, we are talking about large n, right? So I can obviously say that this is not equal to zero because there is always some number n after which this is not equal to zero. So I have the constant 12 over 5, right? 12 fifths. And this thing, I can separately show you that this thing is also convergent to zero. a n plus b divided by 5n squared plus, or minus, doesn't matter really, minus 7n plus 13. So most important, this is n to the first degree and this is n to the second degree. Okay, so what should I do basically? Well, I can just divide by n, so I will have a plus b over n divided by 5n minus 7 plus 13 over n, right? Now the limit of the ratio is the ratio of the limits. The ratio of this is the sum of the limit is equal to the limit of sum, which is constant a. And this thing, you see, 5n, it does not have a limit, it infinitely grows. So whenever I divide, it would be zero as a limit, as 1n, same thing. So this is a constant over 5n, doesn't really matter, it's the same thing. n will grow to infinity and the ratio will go to zero. So basically manipulating this using limit of ratio and limit of sum and other these little theorems, which I have just mentioned to you, we can reduce this to basically the answer. So in this case, this is zero, so my overall limit would be 12 over 5, right? Which are coefficients at the highest degree. So this is just another example of the limits. Now I will address actually these issues a little bit more precise. I will actually prove that if I have two polynomials, then if my denominator has the same degree as denominator, then the limit would be the ratio of their coefficients. Now in all other cases, either my denominator would have a greater degree, in which case the limit is zero, or denominator would have a lower degree, in which case there will be no limit, it will go to infinity. But that's a separate lecture a little bit later. But for today, all I wanted was just to refresh your memory about what are the limits of sequences, and again remind some terminology, convergent, nonconvergent, couple of examples, and properties of the limits like limit of sum is equal to sum of limits, limit of product is equal to product of limits, etc. Okay, I would suggest you to read the notes for this lecture on Unisor.com and try to prove all these theorems which are listed in this lecture without looking at the proof which is presented in the algebra section where I first introduce these properties. So it will be a very nice exercise. That's it, thank you very much and good luck.