Theorem for limits of composite functions: when conditions aren't met | AP Calculus | Khan Academy

In a previous video, we used this theorem to evaluate certain types of composite functions. In this video, we'll do a few more examples that get a little bit more involved.

So let's say we wanted to figure out the limit as x approaches 0 of f of g of x.

First of all, pause this video and think about whether this theorem even applies. Well, the first thing to think about is what is the limit as x approaches 0 of g of x to see if we meet this first condition.

So if we look at g of x right over here, as x approaches 0 from the left, it looks like g is approaching 2. As x approaches zero from the right, it looks like g is approaching two, and so it looks like this is going to be equal to two. So that's a check.

Now let's see the second condition: is f continuous at that limit at two? So when x is equal to two, it does not look like f is continuous. So we do not meet this second condition right over here.

So we can't just directly apply this theorem, but just because you can't apply the theorem does not mean that the limit doesn't necessarily exist.

For example, in this situation, the limit actually does exist. One way to think about it: when x approaches 0 from the left, it looks like g is approaching 2 from above, and so that's going to be the input into f. And so if we are now approaching 2 from above, here's the input into f; it looks like our function is approaching zero.

Then we can go the other way. If we are approaching 0 from the right, right over here, it looks like the value of our function is approaching 2 from below. Now, if we approach 2 from below, it looks like the value of f is approaching 0.

So in both of these scenarios, our value of our function f is approaching 0. So I wasn't able to use this theorem, but I am able to figure out that this is going to be equal to 0.

Now let me give you another example. Let's say we wanted to figure out the limit as x approaches 2 of f of g of x. Pause this video, and we'll first see if this theorem even applies.

Well, we first want to see what is the limit as x approaches 2 of g of x. When we look at approaching 2 from the left, it looks like g is approaching negative 2. When we approach x equals 2 from the right, it looks like g is approaching 0.

So our right and left-hand limits are not the same here, so this thing does not exist. It does not exist. And so we don't meet this condition right over here, so we can't apply the theorem.

But as we've already seen, just because you can't apply the theorem does not mean that the limit does not exist. But if you like pondering things, I encourage you to see that this limit doesn't exist by doing very similar analysis to the one that I did for our first example.