# Proof of special case of l'Hôpital's rule

## Video transcript

What I want to go
over in this video is a special case
of L'Hopital's Rule. And it's a more
constrained version of the general case
we've been looking at. But it's still very powerful
and very applicable. And the reason why we're going
to go over this special case is because its proof is
fairly straightforward and will give you an intuition
for why L'Hopital's Rule works at all. So the special case
of L'Hopital's Rule is a situation where
f of a is equal to 0. f prime of a exists. g of a is equal to 0. g prime of a exists. If these constraints
are met, then the limit, as x approaches a of
f of x over g of x, is going to be equal to f
prime of a over g prime of a. So it's very similar
to the general case. It's little bit
more constrained. We're assuming that
f prime of a exists. We're not just
taking the limit now. We're assuming f prime of a and
g prime of a actually exist. But notice if we substitute
a right over here we get 0/0. But that if the
derivatives exist we could just evaluate
the derivatives at a, and then we get the limit. So this is very close
to the general case of L'Hopital's Rule. Now let's actually prove it. And to prove it, we're going
to start with the right hand and then show that if we use
the definition of derivatives, we get the left hand
right over here. So let me do that. So I'll do it right over here. So f prime of a
is equal to what, by the definition
of derivatives? Well, we could view
that as the limit as x approaches a of f of x
minus f of a over x minus a. So this is literally just
a slope between two points. So like, if you have your
function f of x like this, this is the point a, f
of a right over here. This right over here
is the point x, f of x. This expression
right over here is the slope between
these two points. The change in our y value
is f of x minus f of a. The change in our f
value is x minus a. So this expression is just
the slope of this line. And we're just taking
the-- let me actually do that in a different
color-- the line that connects these two points,
that's the slope of it. I'll do that in white. The slope of the line that
connects those two points. And we're taking
the limit as x gets closer and closer
and closer to a. So this is just
another way of writing the definition of
the derivative. So that's fine. Let's do the same
thing for g prime of a. So f prime of a
over g prime of a, is going to be this
business which is in orange, f prime of a over g prime of a. Which we can write as
the limit as x approaches a of g of x minus g
of a over x minus a. Well, in the numerator,
we're taking the limit as x approaches a, and
in the denominator, we're taking the limit
as x approaches a. So we can just rewrite this. This we can rewrite as
the limit as x approaches a of all this
business in orange. f of x minus f of
a, over x minus a, over all the business in green. g of x minus g of a, all
of that over x minus a. Now, to simplify this, we
can multiply the numerator and the denominator by x minus
a to get rid of these x minus a's. So let's do that. Let's multiply by x
minus a over x minus a. So the numerator, x minus a,
and we're dividing by x minus a. Those cancel out. And then these two cancel out. And we're left with this thing
over here is equal to the limit as x approaches a
of, in the numerator we have f of x minus f of a. And in the denominator, we
have g of x minus g of a. And I think you see
where this is going. What is f(a) equal to? Well, we assumed f
of a is equal to 0. That's why we're
using L'Hopital's Rule from the get go. f of a is equal
to 0, g of a is equal to 0. f of a is equal to 0.
g of a is equal to 0. And this simplifies to the
limit as x approaches a of f prime of x, sorry of f of
x, we've got to be careful. Of f of x over g of x. So we just showed that if f of
a equals 0, g of a equals 0, and these two derivatives exist,
then the derivatives evaluated at a over each other are
going to be equal to the limit as x approaches a of
f of x over g of x. Or the limit as x
approaches a of f of x over g of x is going to
be equal to f prime of a over g prime of a. So fairly straightforward
proof for the special case-- the special case, not
the more general case-- of L'Hopital's Rule.