 In the previous video, we introduced rational functions and talked about vertical asymptotes, things that can happen on the graph that causes infinite rip. It turns out that rational functions can also have horizontal asymptotes, which the idea is going to be like the following that as x goes towards, towards infinity, there might actually be some finite value that the function approaches, right? So as x approaches infinity, you're going to see that y approach is actually something that's not infinity. It's going to be something finite that can happen. This is its in behavior. The same thing can also happen on the left. And so this is what we mean by a horizontal asymptote. The graph will start approaching some horizontal line as you go to the far left and the far right. And this is going to be an approximation of what happens to our function as x gets really, really, really big on the left or it gets really, uh, it goes on the far left or on the far right. Uh, so the, this is really getting measuring the in behavior, the in behavior of our rational function. Now, if you have like a monomial x to the n as x goes towards infinity, it's going to point up and as x goes to negative infinity, you'll point down or point up depending on whether it's even or odd. Uh, the reciprocal function one over x, the end is going to kind of do something similar as x goes towards infinity. Uh, x the end is going to get really, really, really big. And if you have, if you have one over a big number, this is actually equal to a small number. And so this is what's happening here for, for these reciprocal functions. As x goes towards infinity in general, this thing is going to go off towards zero one over x to the end is going to go to, excuse me, as x goes to infinity, one over x to the end, it's going to go towards zero. It's going to get really, really, really small. And we want to talk about these principles a little bit more here. Uh, what type of horizontal asymptotes can we anticipate for a rational function? Well, when the top is bigger than the denominator, we have this improper fraction in that situation. You actually have no horizontal asymptote. Uh, in this situation, you're, you're, you'll have no horizontal asymptote. It'll be kind of, it'll be approximately a polynomial, meaning that it'll go off towards infinity or negative infinity by the same rules we used when we talked about polynomial functions. Uh, this is what we call the top heavy rational function. It's top heavy. If the top is bigger than the bottom. On the other hand, what if the bottom, excuse me, what, excuse me, what if the top is smaller than the bottom? What if the denominator has a bigger degree than the numerator in that situation? That's like the situation we have one over x to the end in that situation. You have a horizontal asymptote y equals zero. This is of course just the x axis in this situation. Your function will be approximately the same thing as one over x to the end for some power. And so this is what we refer to as the bottom heavy case. The final possibility is what if the two degrees are the same, uh, the degree of the top equals the degree on the bottom. We refer to this as the balance case. And in that situation, you do have a horizontal asymptote and the horizontal asymptote will be a number, will be y equals a number P over Q, where P is the number on the leading coefficient on top and Q is the leading coefficient on the bottom. Let me show you some examples to illustrate the, the principle here. So let's say that we have a rational function r of x equals x over, uh, x over x squared minus four. I want you to realize that this right here is going to be our bottom heavy function. It's bottom heavy. We see that because the numerator is degree one. The denominator is degree two. And so the trick I want you to think of here is when you have this bottom heavy, I just want you to look at the leading terms on the top and bottom. That's basically all that matters. Cause as x goes to infinity, only the leading term mattered for polynomials. That same thing is also true for rational functions. So as x goes towards infinity, this will look like approximately x over x squared, which simplifies just to be one over x. And so as x approaches infinity, y will approach zero, right? Cause as x gets really, really big, it's reciprocal get really, really small. As x approaches negative infinity, y will likewise approach zero. And so that tells us there's a horizontal asymptote at y equals zero, aka the x-axis. That's what happens when you have a bottom heavy function. Okay. Uh, let's look at another example. Let's try this one right here. F of x equals two x plus three over x minus one. This is an example of a balanced rational function. It's balanced because the leading terms on top and bottom have the same degree. This thing, if we just look at the leading terms will look like two x over x, which simplifies just to be two or two over one, if you prefer. And so this is going to be our horizontal asymptote as x approaches infinity, y will approach the number two as x approaches negative infinity, y will approach again, the number two. And therefore this function has a horizontal asymptote at the horizontal line y equals two. It's a line. So we need, we don't just say two, it's y equals two. It's the equation of a horizontal line. And then the last, I guess there's, there's two more I want to do. Uh, the next one here, let's do g of x right here. Notice this one is an example of a top heavy rational function. It's top heavy. So if we, if we just look at the leading terms on top and bottom, the top is bigger than the bottom. If we look at the end behavior, this is just an approximation. This will be, as x goes to infinity or negative infinity, this will be approximately the same thing as x cubed over two x squared, which simplifies to be one half x. That's the end behavior. The end behavior of this rational function will be essentially the same thing as one half x. So as x approaches infinity, we're going to see that y approaches infinity as well. And as x approaches negative infinity, we're going to see that y approaches negative infinity as well. So in this situation, the horizontal acetote does not exist. It doesn't have one because it was top heavy. Uh, let's look at another example. Uh, in this situation, though, we see that again, that this thing is balanced. As x goes towards infinity, this will be approximately the same thing as just x squared over x squared, uh, which is, this is just a one in that situation. So as x approaches infinity, y approaches one as x approaches negative infinity, y approaches one again. And so we see that there's this horizontal acetote at the line, y equals one for this example right here, because again, it was a balanced rational function.