a little bit more practice taking limits

of composite functions.

Here, we want to figure out what is the limit

as x approaches negative one 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, what is the limit as x approaches negative one

of h of x, and if that limit exists, then input that into g.

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 one

of h of x does not exist.

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 one 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?

As we approach negative one from the right hand side,

it looks like h is approaching negative two.

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 one from the right.

So it's approaching from values larger than negative two

of g of h of x.

G of h of x.

I'm color coding it to be able to keep track of things.

This is analogous to saying what is the limit,

if you think about it as x approaches negative two

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 can see that g is approaching three.

So this right over here is going to be equal to three.

Now, let's take the limit as x approaches negative one

from the left of g of h of x.

What we could do is first think about what is h approaching

as x approaches negative one from the left?

As x approaches negative one from the left,

it looks like h is approaching negative three.

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.

H of x is approaching negative three from above,

or we could say from values greater than negative three,

and then of g of h of x.

Another way to think about it,

what is the limit as the input to g

approaches negative three from the right?

As we approach negative three from the right,

g is right here at three,

so this is going to be equal to three again.

So notice the right hand limit and the left hand limit

in this case are both equal to three.

So when the right hand and the left hand limit

is equal to the same thing, we know that the limit

is equal to that thing.

This is a pretty cool example,

because the limit of, you could say the internal function

right over here of h of x, did not exist,

but the limit of the composite function still exists.