Limits at infinity of quotients with trig | Limits and continuity | AP Calculus AB | Khan Academy

So let's see if we can figure out what the limit as x approaches infinity of cosine of x over x squared minus one is. And like always, pause this video and see if you can work it out on your own.

Well, there's a couple of ways to tackle this. You could just reason through this and say, well look, this numerator right over here, cosine of x, that's just going to be oscillating between negative 1 and 1.

Cosine of x is going to be greater than or equal to negative 1, or negative 1 is less than or equal to cosine of x, which is less than or equal to 1. So this numerator just oscillates between negative 1 and 1 as x changes, as x increases in this case.

Well, the denominator here, we have an x squared, so as we get larger and larger x values, this is just going to become very, very, very large. So we're going to have something bounded between negative 1 and 1 divided by very, very infinitely large numbers.

And so if you take a bounded numerator and you divide by an infinitely large denominator, well that's going to approach zero. So that's one way you could think about it.

Another way is to make this same argument, but to do it in a little bit more of a mathy way. Because cosine is bounded in this way, we can say that cosine of x over x squared minus 1 is less than or equal to, well the most that this numerator can ever be is 1.

So it's going to be less than or equal to 1 over x squared minus 1. And it's going to be a greater than or equal to, it's going to be greater than or equal to, well the least that this numerator can ever be is going to be negative 1.

So negative 1 over x squared minus 1. And once again, I'm just saying look, cosine of x at most can be 1 and at least is going to be negative 1. So this is going to be true for all x.

And so we can say that also the limit, the limit as x approaches infinity of this is going to be true for all x. So limit as x approaches infinity, limit as x approaches infinity.

Now this here, you could just make the argument look, the top is constant, the bottom just becomes infinitely large. So this is going to approach 0.

So this is going to be 0 is less than or equal to the limit as x approaches infinity of cosine x over x squared minus 1, which is less than or equal to, well this is also going to go to 0. You have a constant numerator and an unbounded denominator, this denominator is going to go to infinity.

And so this is going to be 0 as well. So if our limit has to be between zero, if zero is less than or equal to our limit, is less than or equal to zero, well then this right over here has to be equal to, it has to be equal to zero.