 Welcome back to our lecture series, Math 12-10, Calculus I for students at Southern Utah University. As usual, I'll be your professor today, Dr. Andrew Miseldine. In this lecture 15, I'm actually going to continue the topics of limits at infinity that we introduced in lecture 14 previously in this lecture series. In lecture 14, we introduced the idea of a vertical asymptote and a horizontal asymptote. What I want to do at the start of lecture 15 is to actually try to distinguish the differences between them because it both involve limits at infinity in a manner of speaking, but it's different types of limits at infinity. When one is talking about a vertical asymptote, what they mean is something like the following. We take the limit as x approaches some finite number a. We're either approaching a from the right or approaching a from the left or maybe we're approaching it from both sides doesn't matter. We take the limit as x approaches a from some direction of f of x and this flow limit turns out to be plus or minus infinity. In other words, as x approaches a finite number, plus or minus, we get that y approaches plus or minus infinity. If the y-coordinate is going towards infinity, but the x-coordinate is not, that is what describes a vertical asymptote using the language of limits. Now, on the other hand, a horizontal asymptote is going to coincide in the other direction. That is, if we have a limit as x is approaching plus or minus infinity of f of x, and we end up with some number a, just some number of which we could be approaching it from the right or the left. I don't really care. It doesn't make much of a difference in this context, but that's going to be a horizontal asymptote. As x is approaching plus or minus infinity, y will be approaching a finite number either from above or below. Again, the approach doesn't matter so much in that regard, but that's the difference between a horizontal asymptote and a vertical asymptote. The behavior around a vertical asymptote is when you get close to x equals a, the y-coordinate will explode, either positive or negative infinity. On the other hand, when it comes to a horizontal asymptote, as x goes to the extreme, as x explodes either to the far right or the far left, then the y-coordinate will asymptotically approach some number here. When it comes to vertical asymptotes, most likely the vertical asymptote came into existence because you had some type of division by zero, maybe divided zero from the right or from the left or whatever, and so as such, you get this graph, you have this line that's outside the domain of the function for which the function is approaching positive or negative infinity of that situation. Because of that reason, it's very calm, because the vertical asymptotes typically come from division by zero, which will be outside the domain of the function, our function's not going to touch its vertical asymptote because it's typically outside the domain. That's not a hard rule, but it's a very common practice, especially since most of the type of vertical asymptotes we occur that we see have occurred because of some type of rational expression, like one that we'll consider in just a moment. On the other hand, there is no stipulation that a function cannot touch its asymptote, and in particular, that's very much the case for horizontal asymptotes. You could have a function which asymptotically approaches one side, the other side, and then the middle, it might actually cross, it's horizontal asymptote. That's a very common thing. I don't want you to assume asymptotic means we can't touch it. It just, asymptotic just means that as we go to the extreme, as we go to the ends of the map, whether that's with respect to the domain or the range, as we go to the end, we're going to get closer and closer to this thing, so these things will be approximately the same thing when we take them to the extreme. That's the difference between a vertical and horizontal asymptote, how do you actually compute that? Let's say we're given some rational function like y equals two x plus one over x minus two. If we want to find the vertical asymptotes, I should color code this better. If we want to find the vertical asymptotes, this is going to come about when my denominator is equal to zero. So in this situation, we have to investigate what makes x minus two go to zero? Well, that's going to happen at x equals two. And so then we then look at the limits, take the limit as x approaches two from the right of two x plus one over x minus two, and we're also going to consider the limit as x approaches two from the left of two x plus one and x minus two. So the things we have to consider the left approach and the right approach because these could very well be different. If we plug this into the expression, because after all, rational function is continuous on its domain, we're going to end up with two times two plus one over two plus minus two. Simplify the numerator, we get two times two, which is four plus one is five. And then if we're a little bit bigger than two and we subtract two, we're going to be a little bit bigger than zero. This ends up being positive infinity. On the other hand, if we approach from the left, we're still going to get two times two plus one over two minus minus two, which will give us a five over zero minus because we're a little bit less than two when we subtract two will be a little bit less than zero. This is going to turn out to be negative infinity. You'll notice that in my calculation I did, I only put the two plus and the two minus in the denominator. I actually didn't care about the numerator because we're a little bit bigger than five or a little bit less than five, we're still positive. And when it comes to ratios that end up being infinite, we only care about the sign. Is it positive infinity or negative infinity? And so that's why I paid attention to the sign and the denominator because zero itself is actually a sinless number. It's neither positive nor negative. So I need to focus on the approach. Am I approaching zero from the positive side or approaching zero from the negative side? And so what this tells me about my function, if I actually could have some a lot of information about its graph right now, it turns out if we were to graph the function at X equals two, there's a vertical asymptote, like we said. And so the function would be, it would be approaching infinity from the right-hand side and it would be approaching negative infinity from the left-hand side. We know that already. What about the horizontal asymptote? The horizontal asymptote will be computed by taking the limit as X approaches infinity of two X plus one over X minus two. And as we've seen previously, when it comes to finding limits as you're going towards infinity, only the dominant term is gonna matter in that situation. So you get two X minus one, which is gonna equal the limit as X approaches infinity to X over X. That simplifies just to be the limit as X approaches infinity of two. So this is gonna be a two. And admittedly, there's no difference if we had been approaching negative infinity. So I mean, just like the vertical asymptote, we wanna approach it from the right, we wanna approach it from the left, but notice this calculation wouldn't change. The limit would here would be two. And so this tells us that our function, it had a vertical asymptote at X equals two and it's gonna have a horizontal asymptote likewise at Y equals two. So the number two showed up again, but for a different reason. And so we see this time that as our function goes to the extreme, our function's gonna approach Y equals two on the right. It's gonna do this also from above. That we're gonna see that it comes from above and then the other one's gonna come from below. We can add that extra bit of specification on signs if we need to, if it matters, absolutely. But we can add that if we really need to. So the only thing I would say to finish up this graph is I might be interested in what's the X intercept? What's the Y intercept of this function? Setting X equals to zero, you get the Y intercept is negative one half. When you solve for the numerator equal to zero, you'll see that the X intercept is one half, negative one half, excuse me. So when you piece those things together, we can very well get the whole picture just by using the intercepts and asymptotes of this function. All right, let's look at one that's a little bit more complicated. It will take Y equals X squared minus four over two X squared minus three X minus two. In this setting, it's gonna be very important to try to factor things. We would like this function to be factored. The numerator, it gives you X minus two and X plus two, just a difference of squares. The denominator takes a little bit more effort. You're gonna have to basically do some type of reverse foil method. There's a thousand different names for these things, the rainbow method, the AC method, the reverse foil method, what have you. In the process of factoring, you're gonna end up with an X minus two and a two X plus one, like so. For which you can notice that, hey, there is some type of reduction at X minus two. That actually does tell us something that our graph of the function is gonna have a removed point. It has a removed point at two comma because two is what makes that thing go to zero. How do we figure out what the other coordinate is? Well, that comes from finding the limit as X approaches two of our function, Y here, but you can use the simplified version when you cancel out the X minus two because the original function and the simplified version, they only differ by what happens at X equals two. So we can use the simplified version to calculate the limit X plus two to X plus one for which our limit there, we can just plug in two by continuity. So we get two plus two over two times two plus one. We end up with four fifths. So we do have this removed point at two and four fifths. Okay, the question didn't ask for that, but that is interesting. If we wanna figure out the vertical asymptotes, well, those are gonna be what makes the denominator go to zero. This is actually important to mention. The vertical asymptotes will occur at those places that make the denominator go to zero, but don't make the numerator go to zero in reduced form. So notice X minus two did make the denominator go to zero, but when you plug in X equals two in the original function, you end up with zero over zero. That does not necessarily mean a vertical asymptote. You need to reduce it. And so when you put this in smallest terms, no longer does X minus two make the denominator go to zero. And so in this case, if you set two X plus one equal to zero, two X plus one equal to zero, then you're gonna see that negative one half is a vertical asymptote for this function. Investigating what happens. In terms of limits, we can use the simplified version of this function. It makes life a lot easier. X plus two over two X plus one. Why did I put one half there? That was where the removed point was. Sorry about that. We need to put in negative one half. That's what we care about. If we approach it from the right, then we're gonna end up with the numerator. You're gonna get, well, honestly, we don't really care about the numerator because it doesn't make the numerator doesn't go to zero in this situation, but it'll be what it is. In the denominator, we're gonna get two times negative one half from above plus one. This becomes something in the numerator. Don't really care. The sign does matter. So we don't need to care about that. Two take away a half. It's gonna be one and a half, 1.5. The denominator is gonna be zero plus. So this does turn out to be infinity. And so by a similar calculation, if we plug in negative one half from the left of this X plus two over two X plus one, you're gonna end up with 1.5 over zero from the left. So you get negative infinity. Like, so that's our vertical asymptotes. For the horizontal asymptotes, we then investigate, we take the limit, horizontal asymptote at Y equals. We'll see what it is in a moment. As X approaches infinity. Again, we can use the simplified version for all of our limit calculations. So you get X plus two over two X plus one. Again, this is a balanced rational function. So this is gonna give us a limit of one half. And this will be the same whether we approach positive or negative infinity. The nice thing about rational functions is that if it has a horizontal asymptote, it will be the same as you go towards infinity and negative infinity. We will see in a future video that with other functions, like involving square roots or other functions, that the left horizontal asymptote can be very different than the one on the right. Let us do consider a function which may have a horizontal asymptote. That's not necessarily a rational function. Let's consider the function E to one over X right here. Now, just for simplification, we're gonna take the function E to the X and denote it as EXP for the moment of X. So we can rewrite this whole thing as the limit, as X approaches infinity of EXP of one over X. The reason I wanna do that is to make the composition of functions even more obvious here. We put the rational function of one over X inside of the exponential function EXP. And since EXP, the natural exponential is a continuous function, we can actually bring it out of the limit process. And we're gonna end up with EXP of the limit as X approaches infinity of one over X. And notice that as X approaches infinity, one over X is gonna, it's gonna become zero. So we get EXP of zero, zero from the right, if we're gonna be precise, but that level specificity isn't actually required here because we need to compute E to the zero. And whether you're a little bit above zero, a little bit below zero, the approach isn't gonna change here. We're gonna see this is gonna equal one. So our function, so if we take F of X to equal E to the one over X, we see that it has a horizontal asymptote at Y equals one. If we take the limit as X approaches negative infinity in this situation, you get E to the one over X. The calculation is gonna look very similar. We pull out the natural exponential, we take the limit as X approaches negative infinity right here, gonna get one over X here. And so we end up with getting E to the zero again. Now you're approaching it from below, but again, that distinction doesn't make a difference. And we end up with one. So we see that even for this non-rational function, it has a horizontal asymptote of one on the left side and right side of its graph. Does this function have a vertical asymptote though? I mean, after all, when you look at it, you do see this one over X. And as we've seen, sending the denominator of a rational expression to zero typically causes infinity, right? So it seems like there could be some type of vertical asymptote going on here. So let's investigate what happens. If we take the limit as X approaches zero from the right of E to the one over X, pull out the exponential, we get EXP of the limit as X approaches zero from the right of one over X. This ends up being a positive infinity. So we're gonna get E to the positive infinity, which really, if you take a positive number, that's a number greater than one and raise it to the infinite power, this is gonna be infinity. So that does tell us there's gonna be a vertical asymptote at X equals zero. But notice what happens here when we do the other direction. If we take X as it goes to zero from the left, E to the one over X here, same type of thing, we're gonna pull out the exponential function, we get the limit as one over X as X approaches zero from the left here. This is gonna turn out to be negative infinity this time and therefore we end up with E to the negative infinity. X exponential functions on their left do something very different on the right. On the right-hand side, assuming again the base like E is greater than one, on the right-hand side, these growing exponentials will blow off towards infinity, but on the left-hand side, they actually have a horizontal asymptote. So this turns out to be zero. And so we see that this function has a very, very curious thing happening on its graph. If we were to take just a stab at it, we're just gonna draw the picture ourselves right here. If we look on the right-hand side, the right-hand side, it blows off towards infinity as we approach zero from the right. And as we go off towards in positive infinity, there is this horizontal asymptote at X equals one. So it might get something like this. Excuse me, Y equals one. But on the right-hand side, you also will approach one. You'll do it from below. And then as you get close to zero, let me try that again. When you get close to zero, it actually wants to look like zero itself. It is undefined at zero, so it's gonna be an open point. But we're gonna get a picture of a graph that looks something like this. So this is an example of a function which has a vertical asymptote on one side of the graph, but it doesn't have it on the other side.