Limits at infinity (horizontal asymptotes)
Limits at infinity of quotients (Part 2)
Let's do a few more examples of finding the limit of functions as x approaches infinity or negative infinity. So here I have this crazy function. 9x to the seventh minus 17x to the sixth, plus 15 square roots of x. All of that over 3x to the seventh plus 1,000x to the fifth, minus log base 2 of x. So what's going to happen as x approaches infinity? And the key here, like we've seen in other examples, is just to realize which terms will dominate. So for example, in the numerator, out of these three terms, the 9x to the seventh is going to grow much faster than any of these other terms. So this is the dominating term in the numerator. And in the denominator, 3x to the seventh is going to grow much faster than an x to the fifth term, and definitely much faster than a log base 2 term. So at infinity, as we get closer and closer to infinity, this function is going to be roughly equal to 9x to the seventh over 3x to the seventh. And so we can say, especially since, as we get larger and larger as we get closer and closer to infinity, these two things are going to get closer and closer each other. We could say this limit is going to be the same thing as this limit. Which is going to be equal to the limit as x approaches infinity. Well, we can just cancel out the x to the seventh. So it's going to be 9/3, or just 3. Which is just going to be 3. So that is our limit, as x approaches infinity, in all of this craziness. Now let's do the same with this function over here. Once again, crazy function. We're going to negative infinity. But the same principles apply. Which terms dominate as the absolute value of x get larger and larger and larger? As x gets larger in magnitude. Well, in the numerator, it's the 3x to the third term. In the denominator it's the 6x to the fourth term. So this is going to be the same thing as the limit of 3x to the third over 6x to the fourth, as x approaches negative infinity. And if we simplified this, this is going to be equal to the limit as x approaches negative infinity of 1 over 2x. And what's this going to be? Well, if the denominator, even though it's becoming a larger and larger and larger negative number, it becomes 1 over a very, very large negative number. Which is going to get us pretty darn close to 0. Just as 1 over x, as x approaches negative infinity, gets us close to 0. So this right over here, the horizontal asymptote in this case, is y is equal to 0. And I encourage you to graph it, or try it out with numbers to verify that for yourself. The key realization here is to simplify the problem by just thinking about which terms are going to dominate the rest. Now let's think about this one. What is the limit of this crazy function as x approaches infinity? Well, once again, what are the dominating terms? In the numerator, it's 4x to the fourth, and in the denominator it's 250x to the third. These are the highest degree terms. So this is going to be the same thing as the limit, as x approaches infinity, of 4x to the fourth over 250x to the third. Which is going to be the same thing as the limit of-- let's see, 4, well I could just-- this is going to be the same thing as-- well we could divide two hundred and, well, I'll just leave it like this. It's going to be the limit of 4 over 250. x to the fourth divided by x to the third is just x. Times x, as x approaches infinity. Or we could even say this is going to be 4/250 times the limit, as x approaches infinity of x. Now what's this? What's the limit of x as x approaches infinity? Well, it's just going to keep growing forever. So this is just going to be, this right over here is just going to be infinity. Infinity times some number right over here is going to be infinity. So the limit as x approaches infinity of all of this, it's actually unbounded. It's infinity. And a kind of obvious way of seeing that, right, from the get go, is to realize that the numerator has a fourth degree term. While the highest degree term in the denominator is only a third degree term. So the numerator is going to grow far faster than the denominator. So if the numerator is growing far faster than the denominator, you're going to approach infinity in this case. If the numerator is growing slower than the denominator, if the denominator is growing far faster than the numerator, like this case, you are then approaching 0. So hopefully you find that a little bit useful.