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.