Hey, everyone. When working with vertical asymptotes, we found that they affect the domain of our rational function, and we could even calculate them in a way that's really similar to finding the domain. Now with horizontal asymptotes, they're instead going to affect the range of our rational function because now we're dealing with horizontal lines that go across our y-axis. Now because we don't have a way of finding the range of our function, you may be worried that this is going to get complicated. But don't worry because it literally just depends on only two things that we can determine by just looking at our function, and that's the degree of both our numerator, the top of our rational function, and the denominator, the bottom of our rational function. So with that in mind, let's go ahead and just jump right into finding some horizontal asymptotes. So looking at this first function here, I have f(x) = \frac{1}{x} and we want to be able to determine what our horizontal asymptote is. Now we're going to look at, like I said, the degree of the numerator and the degree of the denominator. So here the degree of my numerator is 0 because there's not even a variable there, and the degree of my denominator is 1. So 0 I know is less than 1, and whenever the degree of your numerator is less than the degree of your denominator, you are always simply going to have a horizontal asymptote at the line y = 0, which is what we can see on our graph is already happening here. Our line or horizontal asymptote is at y = 0. And that's it. So when your degree of your numerator is less than the degree of your denominator, that's all. Now let's look at one other possibility here. So looking at my function here, I have f(x) = \frac{2x + 3}{x}. So looking at the degree of my numerator and denominator, the degree of my numerator is 1, and the degree of my denominator is also 1. I know that 1 is definitely equal to 1. So here the degree of my numerator is equal to the degree of my denominator. Now whenever that happens, I actually need to look at one more thing on my function, and that is my leading coefficients. So I'm going to take the leading coefficient of my numerator and then divide it by the leading coefficient of my denominator, and that will give me my horizontal asymptote. So here the leading coefficient of my numerator is 2. And the leading coefficient of my denominator is this invisible one here. So my horizontal asymptote is 2 divided by 1, which is just the line y = 2, which we can see is exactly where that horizontal asymptote is on our graph. So now that we know how to find them looking at the degrees of our numerator and denominator, let's go ahead and find some horizontal asymptotes of a couple more functions. So looking at this first example here, I have \frac{4x^{2}}{-x^{3} - 5x + 9}. So looking at the degree of our numerator, I have a 2. And then the degree of my denominator I have is 3. So 2 is less than 3. Well, it's definitely less than. So looking back at our original examples, we know that if our numerator is less than the denominator, y = 0, that will give me my horizontal asymptote. So here, I simply have a horizontal asymptote at y = 0, and I'm done. Let's look at one more example. So here I have f(x) = \frac{2x^{2}}{3x^{2} + x - 1}. So here looking at those degrees, I have a degree of 2 in my numerator and I also have a degree of 2 in my denominator. So the degree of my numerator is definitely equal to the degree of my denominator. Now remember, whenever that happens, we need to take a look at one more thing here, our leading coefficients, in order to find our horizontal asymptote. So the leading coefficient of my numerator is 2 and that of my denominator is 3, taking those and dividing them, I end up with a horizontal asymptote at y = \frac{2}{3}. And that's where my horizontal asymptote will be on my graph. And that's all. So we know how to calculate horizontal asymptotes, but I want to mention one more thing that you might see pop up in your studies. So sometimes, whenever we have a horizontal asymptote, the graph of your rational function may actually intersect it and just fully cross it and then approach it from the other side. Now I know that this goes against what we said our definition of asymptotes are, but it's actually just crossing it and then approaching it from the other side. Now you don't need to worry about why this happens or figuring out when it will. You just need to be aware that it can happen. So don't be afraid when you see a graph, and it crosses a horizontal asymptote. It's totally fine. It can happen. It won't always. But just be aware that it can. So we now know how to calculate horizontal asymptotes and how they may appear on our graph. Let's get some practice.