Guillaume De l'Hôpital 1661 - 1704

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

2. Indeterminate forms If we try to evaluate by direct substitution, we get: Consider: or 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.

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

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. not On the other hand, you can apply L’Hôpital’s rule as many times as necessary as long as the fraction is still indeterminate:

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

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

9. Indeterminate Powers 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: Evaluating these forms requires a mathematical trick to change the expression into a ratio.

10. L’Hôpital applied Indeterminate Forms: Example: