Functions with same limit at infinity | Limits and continuity | AP Calculus AB | Khan Academy

The goal of this video is to get an appreciation that you could have many, in fact, you could have an infinite number of functions that have the same limit as X approaches infinity. So, if we were to make the general statement that the limit of some function f of x as X approaches infinity is equal to three, what I want to do in this video is show some examples of that and to show that we can keep creating more and more examples—really an infinite number of examples—where that is going to be true.

So, for example, we could look at this graph over here, and you in other videos will think about why this is the case, but just think about what happens when you have very, very large X's. When you have very, very large X's, the plus five doesn't matter as much. It gets closer and closer to 3x² over x², which is equal to 3. You could see that right over here—it's graphed in this green color. Even when X is equal to 10, we're getting awfully close to three right over there.

Let me zoom out a little bit so you see our axes. So, that is three. Let me draw a dotted line, and the asymptote that is y is equal to 3. You see the function is getting closer and closer as X approaches infinity. But that's not the only function that could do that. As I keep saying, there's an infinite number of functions that could do that.

You could have this somewhat wild function that involves natural logs that, as X approaches infinity, is getting closer and closer to three. It might be getting closer to three at a slightly slower rate than the one in green, but we're talking about infinity. As X approaches infinity, this thing is approaching three.

As we've talked about in other videos, you could even have things that keep oscillating around the asymptote—as long as they're getting closer and closer to it as X gets larger and larger and larger.

So, for example, that function right over there—let me zoom in. When, let's say, X is equal to 14, we can see that they're all approaching three. The purple one is oscillating around it, the other two are approaching three from below. But as we get much larger—let me actually zoom out a ways and then I'll zoom in.

So let's get to really large values. Actually, even 100 isn't even that large if we're thinking about infinity. Even a trillion wouldn't be that large if we're thinking about infinity. But let's go to 200. 200 is much larger than the numbers we've been looking at, and let me zoom in when X is equal to 200. You can see we have to zoom in an awful lot—awful lot—just to see that the graphs still aren't quite stabilized around the asymptote.

They are a little bit different than the asymptote. I am really zoomed in. I mean, look at this scale—each of these are now a hundredth each square. We've gotten much, much, much closer to the asymptote. In fact, the green function—we still can't tell the difference. You can see the calculation is up to three or four decimal places. We're getting awfully close to three now, but we aren't there.

So, the green function got there the fastest—it's an argument. But the whole point of this is to emphasize the fact that there's an infinite number of functions for which you could make the statement that we made: that the limit of the function as X approaches infinity, in this case, we said that limit is going to be equal to three.

I just picked three arbitrarily. This could be true for any function. I'm trying—well, I didn't realize how much I had zoomed in—so let me now go back to the origin where we had our original expression. So there we have it. Maybe I could zoom in this way. So there you have it: the limit of any of these as X approaches infinity is equal to three.