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## AP®︎/College Calculus AB

### Course: AP®︎/College Calculus AB>Unit 1

Lesson 5: Determining limits using algebraic properties of limits: limit properties

# Theorem for limits of composite functions: when conditions aren't met

Suppose we are looking for the limit of the composite function f(g(x)) at x=a. This limit would be equal to the value of f(L), where L is the limit of g(x) at x=a, under two conditions. First, that the limit of g(x) at x=a exists (and if so, let's say it equals L). Second, that f is continuous at x=L. If one of these conditions isn't met, we can't assume the limit is f(L). Created by Sal Khan.

## Video transcript

- [Tutor] 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 zero of f of g of x, 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 zero 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 zero from the left, it looks like g is approaching two, 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 zero from the left, it looks like g is approaching two from above and so that's going to be the input into f and so if we are now approaching two from above here as the input into f, it looks like our function is approaching zero and then we can go the other way. If we are approaching zero from the right, right over here, it looks like the value of our function is approaching two from below. Now if we approach two from below, it looks like the value of f is approaching zero. So in both of these scenarios, our value of our function f is approaching zero. So I wasn't able to use this theorem, but I am able to figure out that this is going to be equal to zero. Now let me give you another example. Let's say we wanted to figure out the limit as x approaches two of f of g of x. Pause this video, we'll first see if this theorem even applies. Well, we first wanna see what is the limit as x approaches two of g of x. When we look at approaching two from the left, it looks like g is approaching negative two. When we approach x equals two from the right, it looks like g is approaching zero. So our right and left hand limits are not the same here, so this thing does not exist, 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.