Guided Practice 2.8: Using derivatives to evaluate limits

Overview

We begin to transition into applications of the derivative in this section, where we learn how derivatives can be used to evaluate seemingly-intractable limit problems such as

The key to this application is a mathematical result called l’Hopital’s Rule, which allows us to replace the expression whose limit we are taking with a simpler expression involving derivatives. Also in this section, we will take up the concept of the infinite limit which examines the behavior of a function as the inputs get very large in both the positive and negative directions.

Learning objectives

Basic objectives: Each student is responsible for gaining proficiency with each of these tasks prior to engaging in class discussions, through the use of the learning resources (below) and through the working of exercises (also below).

• Explain what is meant by a limit that is an “indeterminate form”.
• State L’Hopital’s Rule. Also be able to pronounce the word “L’Hopital” correctly.
• Explain how L’Hopital’s rule is used to calculate limits having an indeterminate form.
• Explain what the expression $\displaystyle{\lim_{x \to \infty} f(x) = L}$ means in plain English.
• State the $\infty$ version of L’Hopital’s Rule.

Advanced objectives: The following objectives are the subject of class discussion and further work; they should be mastered by each student during and following class discussions.

• Use L’Hopital’s Rule to calculate limits that have an indeterminate form 0/0.
• Use L’Hopital’s Rule to calculate limits that have an indeterminate form $\infty/\infty$.

Learning resources

To gain proficiency in the learning objectives, use the following resources. You may include other resources if you wish, in addition to or in replacement of the following.

Textbook: In Active Calculus, read Section 2.8. Make sure to read actively, working through examples and activities as you go.

Video: Watch the following videos at the MTH 201 YouTube playlist (http://bit.ly/GVSUCalculus).

• Quick Review: L’Hopital’s Rule (3:28)
• Using L’Hopital’s Rule with zero (7:27)
• Using L’Hopital’s Rule with infinity (8:46)

Activities

The following activity is to be done during and following your reading and viewing of the resources. Go to student.desmos.com and join using your name in the format Last, First. For example, I would enter my name as Ballif, Serge. Complete each part of the activity. Some of these problems will require you to work them out on paper before entering your answer. Practice producing high quality work so that your work is readable and meaningful. You will receive a mark of Pass if each item response shows a good-faith effort to be right and is submitted prior to the deadline. Remember to use the Piazza discussion board to ask about any questions you have.