L'Hopital's Rule Example 1

Let's say we need to evaluate
the limit as x approaches 0 of

2 sine of x minus sine of 2x,
all of that over x

minus sine of x.

Now, the first thing that I
always try to do when I first

see a limit problem is hey,
what happens if I just try to

evaluate this function
at x is equal to 0?

Maybe nothing crazy happens.

So let's just try it out.

If we try to do x equals
0, what happens?

We get 2 sine of 0, which is 0.

Minus sine of 2 times 0.

Well, that's going to be sine
of 0 again, which is 0.

So our numerator is
going to be equal to 0.

Sine of 0, that's 0.

And then we have another
sine of 0 there.

That's another 0, so all 0's.

And our denominator,
we're going to have

a 0 minus sine of 0.

Well that's also going to be 0.

But we have that indeterminate
form, we have that undefined

0/0 that we talked about
in the last video.

So maybe we can use
L'Hopital's rule here.

In order to use L'Hopital's
rule then the limit as x

approaches 0 of the derivative
of this function over

the derivative of this
function needs to exist.

So let's just apply L'Hopital's
rule and let's just take the

derivative of each of these and
see if we can find the limit.

If we can, then that's going to
be the limit of this thing.

So this thing, assuming that it
exists, is going to be equal to

the limit as x approaches 0 of
the derivative of this

numerator up here.

And so what's the derivative
of the numerator going to be?

I'll do it in a new color.

I'll do it in green.

Well, the derivative of 2
sine of x is 2 cosine of x.

And then, minus-- well,
the derivative of sine

of 2x is 2 cosine of 2x.

So minus 2 cosine of 2x.

Just use the chain rule
there, derivative of

the inside is just 2.

That's the 2 out there.

Derivative of the outside is
cosine of 2x, and we had that

negative number out there.

So that's the derivative of
our numerator, maria, and

what is the Derivative.

of our denominator?

Well, derivative of x is just
1, and derivative of sine

of x is just cosine of x.

So 1 minus cosine of x.

So let's try to
evaluate this limit.

What do we get?

If we put a 0 up here we're
going to get 2 times cosine

of 0, which is 2-- let
me write it like this.

So this is 2 times cosine
of 0, which is 1.

So it's 2 minus 2
cosine of 2 times 0.

Let me write it this way.

Actually, let me just
do it this way.

If we just straight up evaluate
the limit of the numerator and

the denominator, what
are we going to get?

We get 2 cosine of
0, which is 2.

Minus 2 times cosine of--
well, this 2 times 0 is

still going to be 0.

So minus 2 times cosine
of 0, which is 2.

All of that over 1 minus the
cosine of 0, which is 1.

So once again, we get 0/0.

So does this mean that
the limit doesn't exist?

No, it still might exist,
we might just want to do

L'Hopital's rule again.

Let me take the derivative
of that and put it over

the derivative of that.

And then take the limit and
maybe L'Hopital's rule

will help us on the
next [INAUDIBLE].

So let's see if it
gets us anywhere.

So this should be equal to
the limit if L'Hopital's

rule applies here.

We're not 100% sure yet.

This should be equal to the
limit as x approaches 0 of the

derivative of that thing over
the derivative of that thing.

So what's the derivative
of 2 cosine of x?

Well, derivative of cosine
of x is negative sine of x.

So it's negative 2 sine of x.

And then derivative of cosine
of 2x is negative 2 sine of 2x.

So we're going to have this
negative cancel out with the

negative on the negative 2
and then a 2 times the 2.

So it's going to be
plus 4 sine of 2x.

Let me make sure I
did that right.

We have the minus 2 or the
negative 2 on the outside.

Derivative of cosine of 2x
is going to be 2 times

negative sine of x.

So the 2 times 2 is 4.

The negative sine of x
times-- the negative

right there's a plus.

You have a positive sine,
so it's the sine of 2x.

That's the numerator when
you take the derivative.

And the denominator-- this
is just an exercise in

taking derivatives.

What's the derivative
of the denominator?

Derivative of 1 is 0.

And derivative negative
cosine of x is just-- well,

that's just sine of x.

So let's take this limit.

So this is going to be equal
to-- well, immediately if I

take x is equal to 0 in the
denominator, I know that

sine of 0 is just 0.

Let's see what happens
in the numerator.

Negative 2 times sine of 0.

That's going to be 0.

And then plus 4 times
sine of 2 times 0.

Well, that's still sine of 0,
so that's still going to be 0.

So once again, we got
indeterminate form again.

Are we done?

Do we give up?

Do we say that L'Hopital's
rule didn't work?

No, because this could have
been our first limit problem.

And if this is our first limit
problem we say, hey, maybe we

could use L'Hopital's rule
here because we got an

indeterminate form.

Both the numerator and
the denominator approach

0 as x approaches 0.

So let's take the
derivatives again.

This will be equal to--
if the limit exist, the

limit as x approaches 0.

Let's take the derivative
of the numerator.

The derivative of negative
2 sine of x is negative

2 cosine of x.

And then, plus the
derivative of 4 sine of 2x.

Well, it's 2 times
4, which is 8.

Times cosine of 2x.

Derivative of sine of
2x is 2 cosine of 2x.

And that first 2 gets
multiplied by the

4 to get the 8.

And then the derivative of the
denominator, derivative of sine

of x is just cosine of x.

So let's evaluate
this character.

So it looks like we've made
some headway or maybe

L'Hopital's rule stop applying
here because we take the limit

as x approaches 0
of cosine of x.

That is 1.

So we're definitely not going
to get that indeterminate form,

that 0/0 on this iteration.

Let's see what happens
to the numerator.

We get negative 2
times cosine of 0.

Well that's just negative 2
because cosine of 0 is 1.

Plus 8 times cosine of 2x.

Well, if x is 0, so it's going
to be cosine of 0, which is 1.

So it's just going to be an 8.

So negative 2 plus 8.

Well this thing right here,
negative 2 plus 8 is 6.

6 over 1.

This whole thing is equal to 6.

So L'Hopital's rule-- it
applies to this last step.

If this was the problem we were
given and we said, hey, when we

tried to apply the limit we get
the limit as this numerator

approaches 0 is 0.

Limit as this denominator
approaches 0 is 0.

As the derivative of the
numerator over the derivative

of the denominator, that
exists and it equals 6.

So this limit must
be equal to 6.

Well if this limit is equal to
6, by the same argument, this

limit is also going
to be equal to 6.

And by the same argument,
this limit has got to

also be equal to 6.

And we're done.