6.6: L’Hôpital’s Rule
Learning Objectives
A student will be able to:
- Learn how to find the limit of indeterminate form by L’Hospital’s rule.
If the two functions and are both equal to zero at then the limit
cannot be found by directly substituting The reason is because when we substitute the substitution will produce known as an indeterminate form, which is a meaningless expression. To work around this problem, we use L’Hospital’s rule, which enables us to evaluate limits of indeterminate forms.
L’Hospital’s Rule
If , and and exist, where , then
The essence of L’Hospital’s rule is to be able to replace one limit problem with a simpler one. In each of the examples below, we will employ the following three-step process:
- Check that is an indeterminate form To do so, directly substitute into and If you get then you can use L’Hospital’s rule. Otherwise, it cannot be used.
- Differentiate and separately.
- Find If the limit is finite, then it is equal to the original limit .
Example 1:
Find
Solution:
When is substituted, you will get
Therefore L’Hospital’s rule applies:
Example 2:
Find
Solution:
We can see that the limit is when is substituted.
Using L’Hospital’s rule,
Example 3:
Use L’Hospital’s rule to evaluate .
Solution:
Example 4:
Evaluate
Solution:
Example 5:
Evaluate .
Solution:
We can use L’Hospital’s rule since the limit produces the once is substituted. Hence
A broader application of L’Hospital’s rule is when is substituted into the derivatives of the numerator and the denominator but both still equal zero. In this case, a second differentiation is necessary.
Example 6:
Evaluate
Solution:
As you can see, if we apply the limit at this stage the limit is still indeterminate. So we apply L’Hospital’s rule again:
Review Questions
Find the limits.
- If is a nonzero constant and
- Show that
- Use L’Hospital’s rule to find
- Cauchy’s Mean Value Theorem states that if the functions and are continuous on the interval and then there exists a number such that Find all possible values of in the interval that satisfy this property for on the interval
Review Answers
Texas Instruments Resources
In the CK-12 Texas Instruments Calculus FlexBook, there are graphing calculator activities designed to supplement the objectives for some of the lessons in this chapter. See http://www.ck12.org/flexr/chapter/9731.