This document discusses L'Hopital's Rule, which can be used to evaluate limits that produce indeterminate forms like 0/0 or ∞/∞. It explains that L'Hopital's Rule involves taking the derivative of the numerator and denominator and re-evaluating the limit. However, it cautions that the rule can only be applied to quotients that result in indeterminate forms, and certain functions must first be rewritten in quotient form before applying the rule. Examples of limits that can and cannot use L'Hopital's Rule are provided.

1. If the circumstances are right, L’Hopital’s Rule can speed up limits. L’H ôpital’s Rule

2. Recall that there are some functions that when finding limits produce the indeterminate forms 0/0 or ∞/∞. When these come up, they do not guarantee a limit exists, nor what that limit is. Some examples: Rewriting with algebraic techniques can handle some of them, for example dividing out techniques, rationalizing denominators or numerators, or using the squeeze theorem.

3. This extended Mean Value Theorem and L’H ôpital’s Rule are proved in Appendix A at the back of the book. Consider the Mean Value Theorem applied to f(x) and g(x).

4. This rule is used incorrectly if you apply the quotient rule to f(x)/g(x).

5. Let’s look at using L’H ôpital’s Rule on the examples we looked at in the beginning. Direct substitution achieves the indeterminate form 0/0. Letting and Direct substitution achieves the indeterminate form ∞/∞. Let f(x) be the numerator, and g(x) be the denominator.

6. The forms all have been identified as indeterminate. All but the first two would have to be rewritten in quotient form before L’H ô pital’s Rule applies There are similar forms that have been identified as determinate . L’H ô pital’s Rule doesn’t apply. Limit is positive infinity Limit is negative infinity Limit is zero Limit is positive infinity

7. Caution! L’H ô pital’s Rule can be applied ONLY to quotients leading to the indeterminate forms 0/0 or ∞/∞! Find the following limit:

8. Find the following limit:

9. 8.7a p574/ 1,3,5-10, 11,19, 23 8.7b p574/ 25,29,31,39, 51, 57, 76,77,79,81