How to Transform Limit Problems in Order to Use L'Hôpital's Rule - dummies

How to Transform Limit Problems in Order to Use L’Hôpital’s Rule

Although L’Hôpital’s rule is a great shortcut for doing limit problems, you may sometimes be unable to proceed with a limit problem when substitution produces unacceptable forms,

In these instances, you first have to tweak the problem to get an acceptable form before using L’Hôpital’s rule.

For instance, find

Substituting infinity into x gives you

so you’ve got to tweak it:

Now you’ve got the

case, so you’re all set to use L’Hôpital’s rule. The derivative of

and the derivative of ex is ex, so

Here’s another problem: What’s

(Recall that

means that x approaches 0 from the right only; this is a one-sided limit.) First, substitute zero into x (actually, since x is approaching zero from the right, you must imagine plugging a tiny positive number into x, or you can sort of think of it as plugging a “positive” zero into x). Substitution gives you

which results in

one of the unacceptable forms. So tweak the limit expression with some algebra:

Now substitution gives you

so you can finish with L’Hôpital’s rule:

That’s it.