Introduction to limits at infinity | Limits and continuity | AP Calculus AB | Khan Academy

We now have a lot of experience taking limits of a function. So if I'm taking the limit of f of x, we're going to think about what does f of x approach as x approaches some value a. This would be equal to some limit.

Now, everything we've done up till now is where a is a finite value. But when you look at the graph of the function f right over here, you see something interesting happens. As x gets larger and larger, it looks like our function f is getting closer and closer to 2. It looks like we have a horizontal asymptote at y equals 2.

Similarly, as x gets more and more negative, it also seems like we have a horizontal asymptote at y equals 2. So is there some type of notation we can use to think about what is the graph approaching as x gets much larger or as x gets smaller and smaller? The answer there is limits at infinity.

So if we want to think about what is this graph, what is this function approaching as x gets larger and larger, we can think about the limit of f of x as x approaches positive infinity. So that's the notation, and I'm not going to give you the formal definition of this right now. There in future videos we might do that, but it's this idea as x gets larger and larger and larger—does it look like our function is approaching some finite value?

That we have a horizontal asymptote there, and in this situation, it looks like it is. It looks like it's approaching the value 2. For this particular function, the limit of f of x as x approaches negative infinity also looks like it is approaching 2.

This is not always going to be the same. You could have a situation—maybe we had—you could have another function. So let me draw a little horizontal asymptote right over here. You could imagine a function that looks like this. So I'm going to do it like that, and maybe does something wacky like this, and it comes down and it does something like this.

Here, our limit as x approaches infinity is still 2, but our limit as x approaches negative infinity right over here would be negative 2. Of course, there are many situations where, as you approach infinity or negative infinity, you aren't actually approaching some finite value. You don't have a horizontal asymptote, but the whole point of this video is just to make you familiar with this notation.

Limits at infinity, or you could say limits at negative infinity, they have a different formal definition than some of the limits that we've looked at in the past where we're approaching a finite value. But intuitively, they make sense that these are indeed limits.