## Subtitles section Play video

• - [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.

- [Tutor] In a previous video we used this theorem

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

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yukang920108 posted on 2022/07/01
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