Analyzing horizontal asymptotes of rational functions

0:00- [Instructor] What we're gonna do in this video is talk a little bit about horizontal asymptotes. So what are we even talking about? Well, you could imagine some type of a function. And I'm gonna draw the axis here. So that's our y-axis. This is our x-axis. And when we think about horizontal asymptotes, we're thinking about what happens if the function gets very, very positive, or very, very negative.

0:25So another way to think about it, it might do all sorts of things when the x values are small, but as they get large, it might approach some value. And similarly, as they get very negative, it might approach a value here. The way I've drawn it here, our horizontal asymptote, at least the way I drew it, looks like it is the x-axis.

0:47But it isn't always the x-axis. You might have a function with a horizontal asymptote at some other value. That function might look something like this. It might do all sorts of things down here. But then as you get to very large x's, very positive, or very negative x's, you approach that horizontal line, not quite ever get there, and that's why we call it an asymptote.

1:11So let's look at some examples where we're not looking at the graph, and we're just looking at the definition of the function and see if we can figure out the horizontal asymptotes. So I will do this one in pink. So let's say we have f(x) is equal to negative five x squared, minus six x plus four, over two x squared plus eight.

1:34And you might be saying, "How do I figure out the horizontal asymptotes if I'm not graphing it?" Well, the key thing to realize is, what happens when x gets very positive or very negative? Well, when it gets very positive or very negative, the terms that dominate are the highest degree terms.

1:54So for example, we have, this is a rational function. It's expressed as a ratio, or you could say as a fraction in some ways. And you have a polynomial on top, a polynomial on the bottom. But if you look at the highest degree terms, if x becomes very, very, very large, this x squared becomes so much larger than this, or this negative five x squared in magnitude becomes so much larger than this negative six x, or even this four.

2:20And somewhere in the denominator, as x gets larger, either in the positive or the negative direction, that x squared, or the two x squared, is gonna matter a lot more than that eight. So one way to think about horizontal asymptotes is look at the highest degree term in the numerator and the denominator.

2:36And then think about, well what happens if as they get big, you're roughly looking at something like this. Negative five x squared over two x squared, for very positive x's or very negative x's. And in that world, well the x squares would cancel out, and you would approach f(x) is equal to negative five halves.

2:57Or another way, the line, y is equal to negative five halves. And that's true if you put in very negative x's as well, or very positive, the x squared part would cancel out, and you would get two negative five halves. So I'm not saying that this is what this graph looks like, but you could imagine a horizontal line like that at negative five halves.

3:21Let me give you another example. And here I'm gonna have the numerator and the denominator in factored form. So let's say I have g(x) is equal to x minus four squared, over x squared plus five, times three x minus two. So pause this video, have a go at it, before we do this together.

3:44So the key is once again, is to think about the highest degree terms in the numerator and the denominator. And to really help us there, we can just expand out these, we can multiply these things out. So in the numerator here, x minus four squared is x squared minus eight x plus 16.

4:01And then that's going to be over, let's see, x squared times three x, is three x to the third. And then I can have, then I can have x squared times negative two. So minus two x squared, then I could have five times three x, which is 15x, and then I could have five times negative two, which is negative 10.

4:24So what happens when x gets very positive or very negative? Well, the highest degree terms here are the x squared in the numerator, and the three x to the third in the denominator. So this three x to the third, this is going to grow much faster even than that x squared.

4:43So one way to think about it is, as x's get very, very, very large, this is going to be approximately x squared over three x to the third. But then if you think about it that way, well even if you were, since we're approximating for very large x's, for very positive x's, this would be approximately one over three x.

5:06And we know what happens here if x gets very large, that's going to approach y equals zero, Y equals zero. And you could also think about the same thing if you go to very negative values. You are once again going to, because you have a higher degree in the denominator approach y equals zero.

5:24And you might be realizing to do this type of just figuring out the asymptote without even trying to graph the actual function, you just have to look at the highest degree term. So we didn't even have to expand everything out. We could have said, "Oh, the highest degree term in the numerator's gonna be x squared.

5:39Highest degree term in the denominator's gonna be three x to the third, the denominator's gonna grow faster than the numerator. We're gonna approach zero as we go to very positive or very negative x's."