3.7: Indeterminate forms and L’Hospital’s Rule Guillaume De l'Hôpital 1661 - 1704

Indeterminate forms Consider: or If we try to evaluate by direct substitution, we get: Zero divided by zero can not be evaluated. The limit may or may not exist, and is called an indeterminate form. In the case of the first limit, we can evaluate it by factoring and canceling: This method does not work in the case of the second limit.

L’Hospital’s Rule Suppose f and g are differentiable and g’(x) ≠ 0 near a (except possible at a). Suppose that

L’Hospital’s Rule: Examples

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

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

Indeterminate Products This approaches Rewrite as a ratio! This approaches We already know that but if we want to use L’Hôpital’s rule:

Indeterminate Differences This is indeterminate form Rewrite as a ratio! If we find a common denominator and subtract, we get: Now it is in the form L’Hôpital’s rule applied once. Fractions cleared. Still L’Hôpital again. Answer:

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

Indeterminate Forms: Example: L’Hôpital applied