 In this video, I wanted to compute limits of functions that may or may not have horizontal asymptotes. That is, I wanna explore the in behavior of these functions. We've talked about this previously with regard to arithmetic and infinity and in behavior polynomial functions. In this video, we're gonna mostly be focusing on rational functions and identifying what criteria will guarantee they have vertical asymptotes or not. So if you have to take the limit as x approaches infinity of 8x plus six over 3x minus one, it's very tempting just to plug in infinity, right? We could put an infinity for which if we follow some cautious rules, this thing can simplify nicely enough. Eight times infinity with infinity, infinity plus six would be infinity, so that's the top. In the denominator, you get three times infinity, which is infinity, minus one, which would still be infinity. And so we kinda get stumped right here. We get this indeterminate form infinity over infinity. That doesn't tell us a whole lot. We're gonna have to approach this a little bit differently. So one thing we can do when it comes to these type of limits is x approaches infinity here. When you have a rational function, what we can do to try to compute this is to flatten the function. So what I'm saying is, when you take the limit of a monomial as x goes to infinity, you're gonna end up with infinity, which isn't exactly useful in this situation. On the other hand, if you take the limit as x approaches infinity of some reciprocal function one over x to the n, you're gonna get zero in this case, which zero is actually gonna be a lot more beneficial in terms of this limit calculation. So what you're gonna do when you have these rational functions, I want you to identify the leading term, right? Identify the leading term on top and bottom. And I basically want you to kill off the smaller of the leading terms, which in this case, you see 8x and 3x, they're actually balanced. This is what we might call a balanced rational function. The leading terms on top and bottom are the same. They're both x equals one. And so we're gonna times the top and bottom by one over, or one over x to the first, like so. For which case then, when you distribute that thing through, you're gonna get the limit. You're gonna get 8x times one over x, which is an eight. And then you're gonna get six times one over x, which is a six over x in the numerator. In the denominator, we get three x times one over x, which is a three. And then you get negative one times one over x, which is one over, is negative one over x, as x approaches infinity. Now at this moment, plug in infinity and see what happens. You're gonna end up with eight plus six over infinity over three minus one over infinity. And like we mentioned before with our arithmetic with regard to infinity, if you take one over infinity, that's equal to zero. That's a safe calculation, no indeterminate form anymore. And so we end up with eight plus zero over three minus zero. And we end up with eight over three as the limit of this thing. And you'll notice that eight over three, hey, those were the coefficients of the leading terms on top and bottom. And this is something you see when you have a balanced rational function. If you take the coefficient of the numerator and the coefficient of the denominator, their ratio will give you the horizontal asymptote. This function has a horizontal asymptote at y equals eight thirds, the ratio of their leading terms. Let's consider another example. Let's take the limit as x approaches infinity of three x plus two over four x cubed minus one. In this situation, if we identify the leading terms, we're gonna get three x on top of four x cubed on the bottom. And so notice this time it's not balanced. Actually the bottom has bigger. And so we might call this a bottom-heavy rational function. That is the denominator is heavier than the numerator in terms of the degree. So like I said before, we're gonna multiply the ratio by the reciprocal of the smaller leading term. So as we're gonna times top and bottom by one over x. Like so, because if we just plug in infinity right now, we're gonna end up with infinity over infinity again, which isn't worth a lot of beans. Okay, so if we times top and bottom by one over x, we end up with the limit as x approaches infinity of three plus two over x divided by four x squared minus one over x. Now let's plug infinity into this thing. So if we plug in infinity, you're gonna end up with three plus two over infinity divided by four times infinity squared minus one over infinity, for which those division by infinity is gonna give you zero. So you get three plus zero over, well, infinity squared would be infinity times four is infinity. You're gonna get infinity minus zero like so. This simplifies to be three over infinity, which in the end, I would say that's equal to zero. If we wanna be slightly more specific, it's approaching zero from above, but you can just, we'll just say that here is zero. And so this is something that happens with bottom heavy rational functions. We see that it'll have a horizontal asymptote and that horizontal asymptote will be at y equals zero, AKA the x-axis like we saw in this situation right here. Now some people get a little bit upset with this type of example, because it's like, you can't do arithmetic with infinity. That's like dark magic that's been exiled by the cleric or whatever. And so if you wanted to, you could have done the same thing if you take one over x cubed, one over x cubed, that would avoid the arithmetic at infinity entirely. You'll end up with a zero on top and you'll end up with four in the denominator. The reason I didn't do that is that I want you to kind of approach all three of these problems with the same perspective. Take a look at the smaller power that's gonna help you out here. And if you're willing to play around with infinity, which I've taught you how to do that in a previous video, then the strategy will be the same. You wanna kill off the slower growing function, in this case, three x compared to the four x cubed. Another example, and this one considered, the limit is x approaches infinity. Let's look at this rational function, x squared plus two over four x minus three. You're gonna notice in this situation, you get four, you get x squared on top, four x on the bottom. The numerator actually has the faster growing function, the x squared. So one might call this a top heavy function. In which case, then my recommendations do what we did before. You're gonna kill off the smaller power. So we're gonna kill off the one over x here. So this would give us the limit as x approaches infinity of, well, if you distribute that one over x, you're gonna get x squared times one over x, which gives you an x. You'll get two over x, like so. And then in the denominator, you're gonna get a four minus three over x. If you plug in x equals infinity, you're gonna end up with infinity plus two over infinity divided by four minus three over infinity. In which case, then, in the numerator, you're gonna get infinity plus zero. In the denominator, you get four minus zero. You end up with infinity over four, which just becomes infinity. So we see that this function as x approaches zero, we see that y will approach, excuse me, as x approaches infinity, y will approach infinity as well. And so this is what you see with top-heavy functions. They have no horizontal asymptotes. And now it turns out that this function does have what we call an oblique asymptote, which one can find using long division of polynomials, but I won't delve too much into that at the moment. What I wanna do at this slide is summarize what we just saw about horizontal asymptotes of rational functions. So if you have a rational function r of x, that is, it's a quotient of polynomials d of x divided by q of x. If the numerator has bigger degree than the denominator, so this is what we call the top-heavy case, top-heavy, well, then it has no horizontal asymptote, like we just saw in the previous example. The next possibility is if the denominator, excuse me, if the numerator has smaller degree than the denominator, this is what we call the bottom-heavy case. In the bottom-heavy case, then we see that there was a horizontal asymptote, and in fact, it was the x-axis, y equals zero. So as you approach positive infinity or negative infinity, the rational function will approach zero. And then in the last case we talked about, this actually is our first case, we get that the top and bottom have the same degree. In this case, we're a balanced rational function, and as you take the limit as x approaches positive or negative infinity, the rational function will approach the line y equals p over q, where little p is the leading coefficient of capital P, and little q is the leading coefficient of big q. So it's the ratio of leading terms as you see. And so another thing I wanna mention is that these rules, when you put them together, you can actually simplify these calculations in the following way, right? If we looked at our exercises again, I think I'll just go back in the slides. So you look at this one right here, we have this limit of 8x plus six over 3x minus one. One thing I like to mention when it comes to these things, when we talked about a polynomial, like if you didn't have the denominators, like, oh, with the limit as x approaches infinity of 8x plus six, you would just look at the leading term, the dominant term right there. When it comes to a ratio, you're gonna do the same thing, it's just you're gonna take the leading term on top and the dominant term on the bottom, in which case I'm telling you that this limit is gonna be the same thing as 8x over 3x as x approaches infinity. But of course, as you simplify this thing, the x is canceled out and this becomes 8 thirds. For the second example, if we try that approach as well, what if we just look at the dominant terms? You have 3x on top, a 4x cubed on bottom. This would be the same thing as the limit as x approaches infinity of 3x over 4x cubed. As x goes to infinity, the other terms are minuscule in comparison to leading terms. This would simplify to be the, to be 3 over 4x squared, for which as this is a reciprocal function, you can take out the three fourths, take the limit of one over x squared, that's gonna give you zero, like so. And then lastly, if you would do this one right here, we're gonna take the limit of just the leading terms, because as you go towards infinity, nothing else matters. It's kind of like the, you know, the Lincoln Park song. In the end, it doesn't really matter except for the leading terms. That's what the artists forgot to say in their song there. In this case then, you see that the powers would cancel. This becomes, simplifies to become x over 4, and as x approaches infinity, this polynomial will then approach infinity as well. So when it comes to these rational expressions, you just have to look at the ratio of leading terms. You wanna look at the dominant terms on top or bottom, and that'll give you the final expression. Now we can justify why we only have to look at the leading terms, because we can kill them off by multiplying by those reciprocal functions. And you wanna finish this video with one last example to kind of illustrate this idea here. I can see that we have this bottom heavy function that tells me something about horizontal asymptote. Does the function, the natural log of x over x squared plus one have a horizontal asymptote? Well, since the natural log is a continuous function, we can factor out the continuous function from consideration, and this then makes the expression, the limit calculation become the natural log of the limit as x approaches infinity of x over x squared plus one, for which then we wonder what's going on here. This is a bottom heavy function, so I know it's gonna approach zero, but if you forgot, if you don't wanna necessarily delineate between top, heavy, bottom, heavy, whatever, you can just look at the leading terms, right? You take the limit of x over x squared. Those other terms are minuscule in comparison. You can simplify that expression just to be one over x as x approaches infinity right here. And you're gonna end up with the natural log of zero, which technically is undefined. Also to be more specific, if you approach zero from the left, that's not possible for the natural log. If you're a little bit more careful, which we should be here, we're gonna say this is the natural log of zero from the right. So if we approach zero from the right, what happens to the natural log? Now, before anyone flips a biscuit or anything like this, you know, gets extremely upset. When I say things like the natural log of zero from the right, really this is just shorthand for the notation. The limit as x approaches zero from the right of the natural log of x. That's what that's shorthand for. And that is a well-defined limit. This is gonna turn out to be negative infinity. If you approach zero on the natural log, you're gonna get negative infinity. Think of the graph when it comes to that consideration. This tells us that the function, the natural log of x over x squared plus one, does not have a horizontal asymptote. As x approaches infinity, we end up with negative infinity right here. And so that then brings us to the close to lecture 14. In lecture 15 of this series, we're gonna continue with our discussion of limits at infinity. We've talked about limits of polynomial and rational functions. In the next lecture, we're gonna talk about other pre-calculus functions, maybe they involve like radicals, exponentials, trigonometric functions. What tools can we use there? It's a little bit more complicated, but we'll be good to go. By all means, check out those videos there. If you learned something in these videos, give it a like. If you have any questions, post a comment below. I'd love to answer those questions. 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