First Determine That This Is an Indeterminate Form by Directly Substituting in the Limit

L’Hôpital’s Rule

L’Hôpital’s Rule is a very important and simple tool that allows us to find the limit of functions we otherwise could not analytically find the limit to. It helps find the limit of several basic indeterminate forms. The table below lists all the indeterminate forms that apply. In order to find out if the function meets the requirements to use L’Hôpital’s Rule (not all do), “plug in” the limit to each portion of the function (whether it be the numerator/denominator) and see.

The forms above MUST be satisfied to even use the rule. So, make sure you understand how each limit is described by the above indeterminate form before continuing. Once you have established the indeterminate form, you can proceed with the rule.

L’Hôpital’s Rule works like this. If the indeterminate form is either or you can immediately apply the rule by taking the derivative of both the numerator and denominator separately and reevaluate the limit of the quotient. If one of the two indeterminate forms appear again, apply the rule again, else the limit can be found.

Example 1:

Evaluate

First determine that this is an indeterminate form by directly substituting in the limit

Now apply the rule:

Make sense?

Now let’s try the case.

Example 2:

Again determine the indeterminate form:

Now apply the rule:

For the higher order cases, sometimes one derivative is not enough. In example 2, we were able to “cancel” our x’s to find the limit, but if that is not possible, check to see if you still have an indeterminate form, and if so, apply the rule again.

If the problem contains and indeterminate form that is not like examples 1 and 2, we might have to play around with it to achieve a form we can work with. The ultimate goal is to get into one of the forms like examples 1 and 2 (or)

Example 3

Determine

By direction substitution we see we have the form

Therefore, rewrite the limit in another form to achieve one of the two above forms.

All I did was move the e term down to the denominator and reversed the sign.

Now we have the form:

Now we apply the rule:

So the limit is 0.

I will conclude with more example requiring a log change.

Example 4

Find

You should see that this limit fits the case. The , , and all can be solved use logarithms. First assume the limit exists and set it to a variable like y.

Then, take the natural log of both sides.

The last step is doable because the natural log is a continuous function where we are evaluating the limit.

So you see once we found the limit to be 1, we had to go back to original equation relating the limit to y. Recall that we took the natural log of y in the beginning so we must account for that by “e’ing” both sides. This determines the actual limit to be e.

There are several determinate forms that do not require (and cannot use) L’Hôpital’s rule.

They are listed below.