* L'Hôpital's Rule* is a method for computing a
limit of the form

c can be a number, , or . The conditions for applying it are:

1. The functions f and g are differentiable in an open interval containing c. (c may also be an endpoint of the open interval, if the limit is one-sided.)

2. g and are nonzero in the open interval, except possibly at c.

3. is defined, or is , or is .

4. As ,

If these conditions hold, then

In other words, f and g may be replaced by their derivatives.

Note that you're *not* applying the Quotient Rule to .

A proof may be given using some advanced results (e.g. the Extended Mean Value Theorem). I won't give a proof here.

* Example.* Compute

Plugging into gives , so I can apply L'Hôpital's Rule:

* Example.* Compute

Plugging into gives , so I can apply L'Hôpital's Rule:

* Example.* Compute

As , , so I can apply L'Hôpital's Rule:

* Example.* Compute

As , , so I *can't* apply L'Hôpital's Rule. In
fact, since the top and bottom are both positive,

* Example.* Compute

As , (which is *not* 0!). I convert the expression
into a fraction by * rationalizing*:

As , , so I could apply L'Hôpital's Rule. Instead, I'll divide the top and bottom by x:

If you apply L'Hôpital's Rule, and the limit you obtain is undefined, you may not conclude that the original limit is undefined.

* Example.* Compute

As , , so I can apply L'Hôpital's Rule:

The last limit is undefined, because has no limit as . This implies that the 's in the reasoning above aren't valid. When you do a L'Hôpital computation, the equalities are actually provisional, pending the existence of a limit in the chain.

In fact, the original limit exists:

You can handle the indeterminate form by using algebra to convert the expression to a fraction, and then applying L'Hôpital's Rule.

* Example.* Compute

As , . So

As , , so I can apply L'Hôpital's Rule:

The indeterminate form can be handled by taking logs, computing the limit using the techniques above, and finally exponentiating to undo the log.

* Example.* Compute

As , .

Let . Then

So

Therefore,

* Example.* Compute

As , .

Let . Then

So

As , . So convert the expression to a fraction:

As , , so I can apply L'Hôpital's Rule:

That is, . Therefore,

* Example.* Compute .

As , . Set . Take logs and simplify:

Take the limit as , applying L'Hôpital's rule to the fraction:

Hence, .

* Example.* Compute .

This is an indeterminate form . Combine the fractions over a common denominator:

This is an form, so I can apply L'Hôpital's Rule:

Copyright 2018 by Bruce Ikenaga