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## Determining limits using algebraic properties of limits: limit properties

# Limits of composite functions: internal limit doesn't exist

## Video transcript

- [Instructor] All right, let's get 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.

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