End behavior of rational functions
End behavior of rational functions
- [Voiceover] So, we're given this function, f of x, and it equals this rational expression over here and we're asked "What does f of x approach "as x approaches negative infinity?" So, as x becomes more and more and more and more negative, what does f of x approach? And, like always, pause the video and see if you can think about that on your own. Well, one thing that I like to do when I'm trying to consider the behavior of a function as x gets really positive or really negative is to rewrite it. So, f of x, I'm just rewriting it once, is equal to 7x-squared, minus 2x over 15x minus five. Now, an interesting technique to think about what happens to the different terms as x gets very positive or x gets very negative, is to divide both the numerator and the denominator by the highest degree term of x in the denominator. And the highest degree term of x in the denominator is the first-degree term. We have just a single x there. So, let's multiply both the numerator and the denominator by one over x, or another way of thinking about it is we're dividing both the numerator and the denominator by x. And if we're doing the same thing to the numerator and the denominator, if we're multiplying or dividing them by the same value, I should say, well then, I'm just really just multiplying it by one. So, I'm not changing its value. This will make it a little bit more interesting, and a little bit easier for us to think about what happens when x becomes very, very, very negative. So, 7x-squared divided by x, or being multiplied by one over x, is going to be 7x. 2x times one over x, or 2x divided by x, is just two. And then all of that over 15x divided by x, or 15x over x, is just going to be 15. And then you have five over x. Five times one over x is equal to five over x. Minus five over x. Now, this is equivalent, for our purposes, to what we started with but it makes it a little bit easier to think about what happens when x gets very, very, very, very negative. Well, when x gets very, very, very, very, very, very, very negative, this is going to become a very large negative number. You subtract two from it, it really won't matter much. You divide that by 15, well, that's not gonna matter much. And this is just going to become very, very, very small. You're taking five and you're dividing it by ever-larger negative numbers, or more and more negative numbers. So, this right over here is gonna go to zero. This thing over here is gonna go towards infinity. Or, I should say, it's gonna go towards negative infinity. Seven times a negative trillion, seven times a negative googol, seven times a negative googolplex, we're getting more and more negative numbers, this is gonna get, this is going to approach negative infinity. Doesn't matter that you're subtracting two from that. In fact, that'll get even more negative. And it doesn't matter if you then divide that by 15, you're still approaching negative infinity. If you had a arbitrarily negative number, you divide it by 15, you still have an arbitrarily negative number. And, so, you could say that this is going to go to negative infinity. Now, another way that you could've thought about it. This is actually how I do think about it when I'm trying to, when I see these types of problems. I say, well which terms in the numerator and the denominator are going to dominate? And what do I mean by "dominate"? Well, as x gets very positive or x gets very negative, another way to think about it is the magnitude of x gets large, the absolute value of x gets large. The higher degree terms are going to grow much faster than the lesser degree terms. And so, we could say that for large x, for large x, and when I say "large" I mean high absolute value. High absolute value. And if we're going to negative infinity, that's high absolute value. So, f of x is going to be approximately equal to the highest degree term on the top, which is 7x-squared, divided by the highest degree term on the bottom. 15x is going to grow, in fact, this is right over here, this constant. So, as this becomes larger and larger and larger, this is going to matter a lot, lot less. So, it's going to be approximately that. Which is equal to 7x over 15. Well, even here, if you think about what happens when x becomes very, very negative here. Well, you're just gonna get larger, you're gonna get more and more and more negative values for f of x. So, once again, f of x itself is going to approach, is going to go to, negative infinity as x goes to negative infinity. Let's do another one of these. So, here they're telling us to find the horizontal asymptote of q. A horizontal asymptote, you can think about it as what is the function approaching as x becomes, as x approaches infinity, or as x approaches negative infinity. And just as a couple of examples here. It's not necessarily the q of x that we're focused on. But you could imagine a function, let's say it has a horizontal asymptote at y is equal to two, so that's y is equal to two there. Let me draw that line. So, let's say it has a horizontal asymptote like that. Well then the graph could look something like this. It could look, let me draw a couple of them that have horizontal asymptotes. So, maybe it's over here, it does some stuff, but as x gets really large, it starts approaching, the function starts approaching that y equals two without ever quite getting there. And it could do that on this side as well. As x becomes more and more negative. As it gets more negative, it approaches it without ever getting there. Or, it could do something like this. You could have, if it has a vertical asymptote, too, it could look something like this. Where it approaches the horizontal asymptote from below, as x becomes more negative, and from above, as x becomes more positive. Or vice versa. Or vice versa. So, this is just a sense of what a horizontal asymptote is. It'll show you what's the behavior, what value is this function approaching, as x becomes really positive or x becomes really negative. Well, let's just think about it. We could essentially do what we just did in that last example. What happens if we were to, if we were to divide all of these terms by the highest degree term in the denominator? Well, if we divide, so q of x is going to be equal to, the highest degree term in the denominator is x to the ninth power. So, we could say six, 6x to the fifth divided by x to the ninth is going to be six over x to the fourth. And then minus two times x to the ninth. All of that over three over, I'm gonna divide this by x to the ninth, x to the seventh, plus one. Well, if x approaches positive or negative infinity, six divided by arbitrarily large numbers, that's gonna go to zero. Two divided by arbitrarily large numbers, whether they are positive or negative, that's going to go to zero. So your numerator's clearly gonna go to zero. This term of the denominator, three divided by arbitrarily large numbers, whether we're going in the positive or the negative direction, it is gonna approach zero. It'll approach zero from the negative direction, or we could say from below. If we're dealing with very negative x's. If we're dealing with very positive x's, then we're going to approach zero from above. We're gonna get smaller and smaller positive values. So, all of these things go to zero and this right over here is going to be, would stay at, one. And so if you're approaching zero in your numerator and approaching one in your denominator, the whole thing is going to approach zero. So, in the case of q of x, you have a horizontal asymptote at y is equal to zero. I don't know exactly what the graph looks like but we could draw a horizontal line at y equals zero and it would approach it. It would approach it from above or below. Let's do one more. What does f of x approach as x approaches negative infinity? Well, let's divide all of these terms by the highest degree that we see in the denominator. We see an x to the fourth. So, 3x to the fourth divided by x to the fourth is three. Minus seven over x-squared, I'm just dividing by x to the fourth, minus one over x to the fourth, over, x to the fourth divided by x to the fourth is one, minus two over x, plus three x to the fourth. This is an equivalent, this right over here is, for our purposes, for thinking about what's happening on a kind of an end behavior as x approaches negative infinity, this will do. I've just divided everything by x to the fourth. And so what's gonna happen as x approaches negative infinity? This is going to approach zero. This is going to approach zero. This is going to approach zero. And this is going to approach zero. And so, as all of that stuff approaches zero, what we're left with is we're going to approach, we're going to approach three over one, or we could say just three. Another way you could think about doing these is look at the highest degree terms. 3x to the fourth, x to the fourth. Ignore everything else because they're going to be overwhelmed by these higher degree terms. So, you could say f of x is approximately equal to 3x to the fourth over x to the fourth for large magnitude x. Magnitude x. And very negative is still a very large magnitude, large absolute value. And so, 3x to the fourth divided by x to the fourth, f of x is going to be approximately equal to three. Or it's going to approach three. So, that's another way that you could think about it.