In this video, I will introduce the theorem known as L'Hôpital's Rule.
I want to explain what the theorem says, and when it does and doesn't apply.
Then, I will work out some examples in the next video.
L'Hôpital's Rule is a tool that helps us compute limits.
Specifically, let's look at these four examples on the screen.
They are all limits of quotients, and they are all indeterminate forms,
of types zero/zero or infinity/infinity.
They are all examples where, to compute the limit, you cannot just look
at the numerator and denominator separately,
because we always get an indeterminate form.
L'Hôpital's Rule provides use with something we can do to attack all these,
so that we are not stuck.
So, what does this theorem say?
I'm going to begin with two functions, f and g, and a real number a.
I want to compute that the limit as x approaches a
of f(x) divided by g(x).
Well, the theorem actually is more general than that.
The theorem also works for limits at infinity, and limits at minus infinity,
and also for side limits. All those types.
For simplicity, I'm going to write the statement only for limits
as x approaches a number a.
But, the theorem is still true for all those other types of limits.
This theorem says that under certain conditions, which I will go into in a moment,
the limit of f divided by g
is the same as the limit of f' divided by g'.
How would this be helpful?
Normally, the situation is that I want to compute the first limit,
and that's a difficult limit which we don't know how to compute.
So, instead we compute the second limit, which hopefully is easier.
And that happens often, because when we take derivatives of functions,
they simplify a little bit.
Now, what are the conditions for this theorem to work?
The most important one,
(I wouldn't put it first, but it's the most important one)
is that this only works when the limit I am trying to compute,
the limit of of f/g,
is an indeterminate form of type zero/zero or ±infinity/±infinity.
What that means is that for this to work,
either the limit of the numerator and the limit of the denominator
are both zero, separately,
OR each one of them in separately either plus or minus infinity.
Notice that those are the only indeterminate forms of quotients.
So, if I am in any other situation, then I am not allowed to use L'Hôpital's Rule.
But luckily, if I am in any other situation, I don't need L'Hôpital's Rule,
because I wouldn't have an indeterminate form, and then I would be done.
I would have no doubt about what the limit was.
So, most importantly, this rule only applies when I get an indeterminate form:
zero/zero or ±infinity/±infinity.
Apart from this, there are two technical conditions
for the rule to even make sense.
First, I need to require f and g to be differentiable as x approaches a.
Otherwise, it makes no sense to take their derivatives.
When I say "as x approaches a,"
I mean when x is close to a, but not necessarily at a.
It doesn't matter if they are not differentiable at a.
I am also going to require that g and g' are never zero as x approaches a.
So again, when x is close to a, but not necessarily a.
Otherwise, I cannot even write the function f/g, or f'/g',
when x is close to a, because it would be dividing by zero.
If I cannot write those functions, then I cannot make sense of the limits.
So those are technical conditions,
without which the question is not even worth positing.
But, there is a fourth condition, which we often forget
because it doesn't come into play often, but which is very important.
For this theorem to be true, I also need to request
that the second limit, the limit of f'/g'
either exists, is infinity, or is minus infinity.
Otherwise, the theorem does not apply.
Let me elaborate on this. As I said, this is something we often forget.
This means that if the second limit, the limit of f'/g', exists,
then I can conclude that the limit of f/g also exists, and is equal.
If the limit of f'/g' is infinity,
then I can conclude that the limit of f/g is infinity as well.
If the limit of f'/g' is minus infinity,
then I can conclude that the limit of f/g is minus infinity.
But!
If the limit of f'/g' does not exist for a different reason,
say, it is some function that oscillates in a weird way,
and that's why the limit doesn't exist,
then I can draw no conclusion about the original limit!
The original limit may well exist, and be anything...or perhaps it doesn't exist.
So, for the theorem to apply, I need the second limit to either exist,
or be plus or minus infinity.
That's it! That's the theorem.
What about the proof? Well, the proof is rather long.
This is not just one theorem... it is a lot of theorems.
It's a separate theorem for 0/0 and for infinity/infinity,
and there is also a separate theorem for limits at a point a (a real number),
and at infinity, or minus infinity, or side limits...there are a lot of them.
We normally prove one of them from scratch
and then reduce the rest of the versions to it.
The proof of this theorem relies on the Mean Value Theorem,
or rather, on a variant of the Mean Value Theorem.
I'm not going to present the proof here.
If you are interested, it appears in every calculus textbook.
Instead, I would rather focus on presenting examples
and showing how to use this to actually compute limits.
So, in the next video, I will work out a few examples from beginning to end,
to see how to compute limits with this theorem.
In the one after that, I will present an example on when
the pesky Condition 4 comes into play:
a situation where we are not allowed to use L'Hôpital's Rule,
but if we do not notice, and use it anyway, we get a wrong result.