4.4 L’Hôpital’s Rule

Zero divided by zero can not be evaluated, and is an example of indeterminate form. Consider: If we direct substitution, we get: In this case, we can evaluate this limit by factoring and canceling:

Indeterminate Forms

L’Hôpital’s Rule: If is indeterminate or, then:

We can confirm L’Hôpital’s rule by working backwards, and using the definition of derivative:

If it’s no longer indeterminate, then STOP! If we try to continue with L’Hôpital’s rule: which is wrong, wrong, wrong! Example

On the other hand, you can apply L’Hôpital’s rule as many times as necessary as long as the fraction is still indeterminate: not (Rewritten in exponential form.) Example

This approaches Example

Now it is in the form This is indeterminate form L’Hôpital’s rule applied once. Fractions cleared. Still L’Hôpital again.Example

Indeterminate Forms: Evaluating these forms requires a mathematical trick to change the expression into a fraction. When we take the log of an exponential function, the exponent can be moved out front. We can then write the expression as a fraction, which allows us to use L’Hôpital’s rule. We can take the log of the function as long as we exponentiate at the same time. Then move the limit notation outside of the log.

L’Hôpital appliedExample