8.7 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:

What makes an expression indeterminate? Consider: We can hold one part of the expression constant: There are conflicting trends here. The actual limit will depend on the rates at which the numerator and denominator approach infinity, so we say that an expression in this form is indeterminate. Consider:

Finally, here is an expression that looks like it might be indeterminate : Consider: We can hold one part of the expression constant: The limit is zero any way you look at it, so the expression is not indeterminate.

Indeterminate Forms

As becomes: The limit is the ratio of the numerator over the denominator as x approaches 2.

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.)

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