1. 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.
Guillaume De l'Hôpital
Greg Kelly, Hanford High School, Richland, Washington

2. 8.1: L’Hôpital’s Rule
Johann Bernoulli

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

4. L’Hôpital’s Rule:
* of the form 0/0
If is indeterminate*, then:

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

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

7. On the other hand, you can apply L’Hôpital’s rule as many times as necessary as long as the fraction is still indeterminate:
(Rewritten in exponential form.)
not

8. 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

9. 1

10. Indeterminate Form (0): make it look like 0/0 or /
This approaches
This approaches

11. Some other examples:

12. Indeterminate Forms 1, 00, 0: use logarithms

13. Other examples:
Ans: 1
Ans: 1

14. Indeterminate form : combine the terms
This is indeterminate form
Now it is in the form
L’Hôpital’s rule applied once.
Fractions cleared. Still

15. L’Hôpital again.

16. NOT Indeterminate Forms: