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

2. Zero divided by zero can not be evaluated. The limit may or may not exist, and is called an indeterminate form. Consider: or If we try to evaluate by direct substitution, we get: 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. Indeterminate forms

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

4. L’Hospital’s Rule: Examples

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

6. 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

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

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

9. 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. 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. Indeterminate Powers

10. Indeterminate Forms: L’Hôpital applied Example: