Additional Resources
OpenStax Online Textbook
How to Read a Math Textbook
Additional Instructional Resources
Indeterminate Forms and L'Hopitals Rule- L5
Objectives
- Recognize when to apply L’Hôpital’s rule.
- Identify indeterminate forms produced by quotients, products, subtractions, and powers, and apply L’Hôpital’s rule in each case.
- Describe the relative growth rates of functions.
Summary
Derivatives can be used to help us evaluate indeterminate limits of the form through L'Hôpital's Rule, by replacing the functions in the numerator and denominator with their tangent line approximations. In particular, if and and are differentiable at L'Hôpital's Rule tells us that
When we write this means that is increasing without bound. Thus, means that we can make as close to as we like by choosing to be sufficiently large. Similarly, means that we can make as large as we like by choosing sufficiently close to
A version of L'Hôpital's Rule also helps us evaluate indeterminate limits of the form If and are differentiable and both approach zero or both approach as (where is allowed to be ), then