Limits of composite functions: internal limit doesn't exist | AP Calculus | Khan Academy

All right, let's get a little more practice taking limits of composite functions. So here, we want to figure out what is the limit as x approaches negative 1 of g of h of x. The function g we see it defined graphically here on the left, and the function h we see it defined graphically here on the right. Pause this video and have a go at this.

All right, now your first temptation might be to say, "All right, well, what is the limit as x approaches negative 1 of h of x?" And if that limit exists, then input that into g. So, if you take the limit as x approaches negative one of h of x, you see that you have a different limit as you approach from the right than when you approach from the left. So your temptation might be to give up at this point.

But what we'll do in this video is to realize that this composite limit actually exists, even though the limit as x approaches negative 1 of h of x does not exist. So how do we figure this out? Well, what we could do is take right-handed and left-handed limits. Let's first figure out what is the limit as x approaches negative 1 from the right-hand side of g of h of x.

Well, to think about that, what is the limit of h as x approaches negative one from the right-hand side? So as we approach negative 1 from the right-hand side, it looks like h is approaching negative 2. So another way to think about it is this is going to be equal to the limit as h of x approaches negative two.

And what direction is it approaching negative two from? Well, it's approaching negative two from values larger than negative two. H of x is decreasing down to negative two as x approaches negative 1 from the right, so it's approaching from values larger than negative 2 of g of h of x.

I'm color coding it to be able to keep track of things, and so this is analogous to saying what is the limit as... if you think about it as x approaches negative 2 from the positive direction of g. Here, h is just the input into g, so the input into g is approaching negative two from above, from the right, I should say, from values larger than negative two.

And we could see that g is approaching three. So this right over here is going to be equal to 3. Now, let's take the limit as x approaches negative 1 from the left of g of h of x.

So what we could do is first think about, well, what is h approaching as x approaches negative 1 from the left? So as x approaches negative 1 from the left, it looks like h is approaching negative three. So we could say this is the limit as h of x is approaching negative three, and it is approaching negative three from values greater than negative three.

It's going h of x is approaching negative 3 from above, or we could say from values greater than negative 3, and then of g of h of x. So another way to think about it, what is the limit as the input to g approaches negative three from the right?

So as we approach negative three from the right, g is right here at three, and so this is going to be equal to three again. And so notice the right-hand limit and the left-hand limit in this case are both equal to three.

And so, on the right-hand and the left-hand limit, it is equal to the same thing. We know that the limit is equal to that thing. And this is a pretty cool example because the limit of, you could say, I guess the internal function right over here of h of x did not exist, but the limit of the composited function still exists.