1. Guillaume De l'Hôpital 1661 - 1704 8.1: L’Hôpital’s Rule Actually, L’Hôpital’s Rule was developed by his teacher Johann Bernoulli. De l’Hôpital paid Bernoulli for private lessons, and then published the first Calculus book based on those lessons. Greg Kelly, Hanford High School, Richland, Washington

2. Johann Bernoulli 1667 - 1748 8.1: L’Hôpital’s Rule

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

4. If we zoom in far enough, what will the curves begin to look like? The limit is the ratio of the numerator over the denominator as x approaches 2. Straight lines.

5. At x = 2 approaches approaches its tangent line = f (2) g (2) = So how can we use this to help us find the limit here? its tangent line

6. L’Hôpital’s Rule: If is indeterminate, then:

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

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

9. 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.)

10. L’Hôpital’s rule can be used to evaluate other indeterminate forms besides. The following are also considered indeterminate: The first one,, can be evaluated just like. The others must be changed to fractions first.

11. This approaches We already know that but if we want to use L’Hôpital’s rule:

12. 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

13. L’Hôpital again.

14. Indeterminate Forms: Evaluating these forms requires a mathematical trick to change the expression into a fraction. L’Hôpital applied 