1. L’Hopital’s Rule
Section 8.1a

2. Indeterminate Form 0/0 Consider the limit:
If both functions are zero at x = a, then substitution
produces the result 0/0. This is a meaningless
expression that is known as an indeterminate form.
Past experiences with indeterminate forms:
Our new rule will enable us to use our knowledge of
derivatives to evaluate limits that otherwise lead to
indeterminate forms!

3. L’Hopital’s Rule (First Form)
Start with the ratio of the derivatives of the functions:
Def. of Deriv.
Prop. of Limits
The original
limit!!!
Rewrite
Given info.

4. L’Hopital’s Rule (First Form)
Suppose that , that
and exist, and that Then

5. L’Hopital’s Rule (First Form)
Evaluate
Form
Look familiar?

6. L’Hopital’s Rule (First Form)
Evaluate
Form
Verify graphically?

7. L’Hopital’s Rule (First Form)
Evaluate
Form
Verify graphically?

8. L’Hopital’s Rule (Stronger Form)
Sometimes after differentiation the new numerator and
denominator both equal zero at x = a. In such cases we
apply the stronger form:
Suppose that , that and
are differentiable on an open interval
containing , and that on if
Then

9. L’Hopital’s Rule (Stronger Form)
Evaluate
Form
Still
Differentiate again…
No longer
indeterminate
Verify
graphically?

10. L’Hopital’s Rule (One-Sided Limits)
Evaluate and
Both form
What is suggested from the graph of the function?

11. Other Indeterminate Forms
The rule applies to quotients that lead to the indeterminate
forms or In some other cases, it is possible to
use algebra to get to one of these forms…
Evaluate
Form

12. Other Indeterminate Forms
Evaluate
Form
(Remember, we are not truly “substituting in infinity,”
just as these forms are not truly numbers… they are
simply descriptions of function behavior)

13. Other Indeterminate Forms
Evaluate
Form
The graph suggests that the limit exists, so rewrite:
Let

14. Other Indeterminate Forms
Evaluate
Form
The graph suggests that the limit exists, so rewrite:
Now form

15. Other Indeterminate Forms
Evaluate
Form
Still