 Hi, I'm Zor. Welcome to Neuser Education. I would like to conclude the story about the limits of the sequences with one particular kind of sequences, which occurs quite often in the future material which I'm going to present. These are sequences of ratio of two polynomials. Now, this lecture actually is the final within limits for sequences, and the next one would be, I'll talk about the limits of the functions, and that would be basically an introduction to derivatives. This lecture is part of the whole course, which I do recommend you to take in its entirety on Unizor.com. It's much better to go to this website rather than listen to this lecture from YouTube or any other source, because the website contains, first of all, very detailed notes for each lecture, and secondly, you can be a registered student, and you can take exams, which is very, very important. The site is free, so basically you can do it as many times as you want to take any kind of exam, until you will feel comfortable. Alright, so back to the limits, and again, a particular case of the limits for sequences, each member of which is a ratio of two polynomials. Okay, so here is what I actually mean. Consider we have one polynomial, which is basically a function of n, where n is order number, right? And it's polynomial, which means it will look like this. So a are coefficients, n is basically the order number, and this is polynomial, where p is decreasing powers. We are raising our n. And then we have another polynomial, similar one, with different powers. This is b, not a. Okay, so this is polynomial of n of the power p, and this is the polynomial of n power q. Obviously a0 and b0 are not equal to 0, otherwise our polynomial would be a different power. So a0 and b0 are not equal to 0. Now, what we are talking about is a sequence, which can be represented as a ratio of these polynomials. And by the way, in the previous lecture where I was talking about certain examples of indeterminate forms of the limits, I did use a couple of polynomials as examples. All right, so how can we approach this? All right, so basically it's quite simple. Here is what I'm proposing to do. I will write p of n factoring out n to the power of p. Now n to the power of p, I presume, again, p is the highest power, right? So I am factoring out the p and n to the power of p. So what remains? From this member, I will have a0. From this member, I will have a1n to the power p minus 1 minus p, which is minus 1. And the last two members would be ap minus 1n to the power of 1 minus p, or minus p plus 1. And the last one would be n to the power of minus p. And q of n, I will do exactly the same thing. And the highest power is n to the q, right? Power of q. And the remaining would be b0, b1n to the power of minus 1 plus b, well, et cetera. I forgot to put et cetera here. And the last couple of members would be q minus 1. Again, my bet is b and n to the power minus q plus 1. And the last one would be bqn to the power minus q. Okay. Now, why did I do it? Well, for a very, very simple reason, because now my p to the n over q to the n is equal to... I will divide n to the power of p by n to the power of q. I will get n to the power of p minus q. And then I have a ratio... Now, why is it better? For a very simple reason. Each one of these, and there are only p of them, is infinitesimal. Each one of these, and there are only q of them, is infinitesimal. Which means that the whole thing is a sequence which converges to a0 over b0, right? Because all these are converging to 0. And there is only a finite number of these, p of these and q of these. So the limit of the numerator is a0. Limit of denominator is b0. And m is equal to 0, as we have agreed in the very beginning. So the ratio is the limit. Now, so this thing is a limit. It's a converging sequence, converging to this concrete real number. b0 is not equal to 0, so everything is fine. Now, what about this one? Well, this is n to some power. We know what happens with n to some power. If this power is positive, it grows, infinitely grows. If power is equal to 0, it's basically a constant 1. And if power is negative, then the whole thing is diminishing down to 0, right? So it's infinitesimal. Which means that I can actually have the rule here. If p greater than q, so the power of the numerator, the power of the polynomial, which is in the numerator, is greater than 1, which is in denominator, then I have an infinitely growing. There is a little detail here, because it's infinitely growing. The question is, it's growing by absolute value, but depending on the sign of this, basically depending on the sign of a0 divided by b0, it will be either plus infinity or minus infinity, so infinitely growing by absolute value. So it might actually infinitely decreasing to minus infinity or infinitely increasing to plus infinity. Now, if p is equal to q, then the whole thing converges to this number, because this is equal to 1, a constant. And finally, if p is less than q, so the polynomial in the numerator is of a lesser degree than denominator. Then the whole thing is infinitesimal. Infinitesimal. So that's the verdict. You don't really have to pay attention to all these coefficients, which are with n to a lesser powers. The highest powers, these are all you need to know to determine whether there is or there is no limit, and if there is, which one is it? So if n is equal to q, then the result limit and it's a0 over b0. Now, if p is not equal to q, then it's either infinitely growing by absolute value to plus or minus infinity, or it's infinitesimal when it goes to zero. Well, that's it about polynomials, about true polynomials. Now I have a little more advanced, if you wish, topic. What if it's not a polynomial? A polynomial means this is integer, this is integer minus 1, minus 2, et cetera, down to zero. What if I have any kind of powers? So let me just do it again, and I will give you an example of what I need, any powers. An example is simple. For instance, I have n squared plus square root of n divided by n to the power of, let's say, 3 plus 3n squared plus cubic of n plus 1, something like this. So not only integer powers, because this is actually what? It's n to the power of 1 half, and this is n to the power of 1 third. So what if I have any kind of power functions of n connected with pluses or minuses, with some coefficients, whatever coefficients are. So let me just write it down in some general notation. So I have a sequence of two functions of n ratio of n. Now, the function u of n is, it's an additional subtraction of members. Each one of them is some kind of a coefficient, and n to some kind of power. And power can be any, it can be integer, it can be rational, or it can be irrational. n to the power of pi, for instance, whatever, I mean any kind of fantasy. And the last one will be apn to the power of p, with some kind of maximum m to the power of pm. So I have m, well, actually m plus 1, members here. Each member is some kind of coefficient, and n to some kind of a power, and power can be anything. And let's just agree from the very beginning that p0 is maximum among all these powers. That's very important. The same way as with polynomial, when in the first place I had the n to the maximum power. Because everything depends on the n to the maximum power. So here also, this is maximum power, I suppose. Now, same thing, v of n would be, but you know what, since I'm using u, I will use u here, if you don't mind. Doesn't really matter. Coefficient, just to differentiate from the previous case. v0, n to the power, let's say, q0, plus v1, n to the power of q1, plus, etc., plus un. So this sum contains n members. n to the power of p, q, so qn. And again, I assume that same thing as p0 is maximum of pis. I assume q0 is maximum of all q. So this is the top member, top member in the top power. And I will do exactly the same as in the previous case when we were talking about polynomials. Namely, I will factor out n to the power of p0, the maximum. So u of n would be equal to n to the power of p0. And what do I have here? u0 plus whatever else, all other members, which contain, each one of them will n in some kind of a negative less than zero power. I have assumed that p0 is maximum, which means p1 is smaller, p2 is smaller, pm is smaller. So if I factor out p0, whatever is left would be, in this case, p1 minus p0, which is negative because p0 is greater. And everything else will be negative. So all these would be some coefficients and n to some negative powers and only finite number of them, m, which means this would be my infinitesimal. Same thing with vn. It would be v0 from here, plus whatever else would be n to the power, which is negative. Many members, actually n members, each of them contains some coefficient n to the negative power. And again, the same thing as before. As soon as you know that the power is negative and n goes to infinity, it means this is infinitesimal. Finite number of infinitesimals would add up into infinitesimal, which means that this would be my conversion to u1 sequence and this will be conversion to the zero sequence. And when I divide one by another, I will have n to the power of p0 minus q0 times u0 divided by v0 plus some kind of alpha infinitesimal because the ratio would be convergent to u0 plus v0, u0 divided by v0. So if it's convergent, this would plus alpha, which is infinitesimal, variable. And now we can make exactly the same conclusion. If p0 greater than q0, this is the constant, even plus infinitely small, but it's still a bounded variable, the variable which has a limit as bounded as we know. So, and this will be infinitely increasing. If p0 is greater than q0, so the whole thing would be infinitely increasing. So if this highest power is greater than this highest power, we will have infinitely increasing sequence. If they're equal to each other, then their difference is zero and the power of zero is one. So the whole thing would converge to u0 over v0, the main coefficients. And finally, if p0 is less than q0, this is the negative power. So n to the negative power would be infinitesimal sequence times constant to infinitesimal. So, again, as before, all we need in the case we have not necessarily polynomial kind of expressions, but any kind of a power functions of n connected with pluses or minuses with coefficients in the numerator and denominator, which is a pretty wide class of functions. So for all these, all you need to know is the highest power of n and its coefficient in the numerator and the highest power of n with its coefficient in the denominator. And if the highest powers are the same, then you do have a limit. It's the result of the division between coefficients at the highest power. If they're different, then you obviously have, if p0 is greater than q0, you have infinity as a limit by absolute value. So it's plus infinity or minus infinity, but absolute value would be infinity. Depends on the sign of u0 and v0. And in case p0 is less than q0, then you will have infinitesimal value, infinitesimal sequence u divided by v. Okay, basically that's it. We have talked about one particular class of sequences which can be represented either in a relatively simpler case as ratio of two polynomials or it's in a little bit more complicated case as ratio of two functions, each one of them is basically sum of n to sum power. So as long as you have that, you can always, just by examining the members with highest powers in the numerator and denominator, you can actually answer what's the limit just looking at these two main members of this ratio. Okay, that's it for today. Thank you very much and good luck.