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Sal analyzes the limits at infinity of three different rational functions. He finds there are three general cases of how the limits behave. Created by Sal Khan.
Video transcript
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, [reads function] 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^7 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^7 is going to grow much faster than an x^5 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^7 over 3x^7, 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 to each other, we can 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^7's, 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 of 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 gets larger and larger and larger, as x gets larger in magnitude. Well in the numerator, it's the 3x^3 term, in the denominator it's the 6x^4 term, so this is going to be the same thing as the limit of 3x^3 over 6x^4 as x approaches negative infinity. And if we simplify 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 negative number, it becomes one over a very, very large negative number, which is going to get us pretty darn close to zero, just as one over x as x approaches negative infinity gets us close to zero. 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^4, in the denominator it's 250x^3, these are the highest degree terms. So this is going to be the same thing as the limit as x approaches infinity of 4x^4 over 250x^3, 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 just divide 250 - well I'll just leave it like this. It's going to be the limit of 4 over 250 - x^4 divided by x^3 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, and infinity times some number right over here is going to be infinity, so the limit as x approaches infinity of all of this is 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 denomiator, you're going to approach infinity in this case. If the numerator is growing far slower than the denominator, if the denominator is growing far faster than the numerator, like this case, you are then approaching zero. So hopefully you find that a little bit useful.