- [Tutor] In this video, we're going to try to understand
limits of composite functions, or at least a way
of thinking about limits of composite functions
and in particular, we're gonna think about the case
where we're trying to find the limit as x approaches a,
of f of g of x
and we're going to see under certain circumstances,
this is going to be equal to f of the limit,
the limit as x approaches a of g of x
and what are those circumstances you are asking?
Well, this is going to be true
if and only if two things are true,
first of all, this limit needs to exist.
So the limit as x approaches a of g of x needs to exist,
so that needs to exist and then on top of that,
the function f needs to be continuous at this point
and f continuous at L.
So let's look at some examples
and see if we can apply this idea
or see if we can't apply it.
So here I have two functions,
that are graphically represented right over here,
let me make sure I have enough space for them
and what we see on the left-hand side is our function f
and what we see on the right-hand side is our function g.
So first let's figure out what is the limit
as x approaches negative three
of f of g of x.
Pause this video and see,
first of all, does this theorem apply?
And if it does apply, what is this limit?
So the first thing we need to see
is does this theorem apply?
So first of all, if we were to find the limit
as x approaches negative three of g of x, what is that?
Well, when we're approaching negative three from the right,
it looks like our function is actually at three
and it looks like when we're approaching negative three
from the left, it looks like our function is at three.
So it looks like this limit is three,
even though the value g of negative three is negative two,
but it's a point discontinuity.
As we approach it from either side,
the value of the function is at three.
So this thing is going to be three,
so it exists, so we meet that first condition
and then the second question is is our function f
continuous at this limit, continuous at three?
So when x equals three, yeah, it looks like at that point,
our function is definitely continuous
and so we could say that this limit
is going to be the same thing
as this equals f of the limit
as x approaches negative three of g of x,
close the parentheses
and we know that this is equal to three
and we know that f of three
is going to be equal to negative one.
So this met the conditions for this theorem
and we were able to use the theorem
to actually solve this limit.