We use derivatives to give us a “short-cut” for computing limits.

Derivatives allow us to take problems that were once difficult to solve and convert them to problems that are easier to solve. Let us consider L’Hôpital’s rule:

This theorem is somewhat difficult to prove, in part because it incorporates so many different possibilities, so we will not prove it here.

L’Hôpital’s rule allows us to investigate limits of indeterminate form.

Basic indeterminant forms

Our first example is the computation of a limit that was somewhat difficult before.

Our next set of examples will run through the remaining indeterminate forms one is likely to encounter.

Indeterminate forms involving subtraction

There are two basic cases here, we’ll do an example of each.

Sometimes one must be slightly more clever.

Exponential Indeterminate Forms

There is a standard trick for dealing with the indeterminate forms Given and such that falls into one of the categories described above, rewrite as and then we rewrite as Recall that the exponential function is continuous. Therefore Now we will focus on the limit using L’Hôpital’s rule. We will only give a single example.