Guided Practice 1.7: Limits, continuity, and differentiability

Overview

This section introduces and in some cases refines some important conceptual notions related to the derivative. We first introduce the concept of a one-sided limit, which is just a limit where we only approach the target point from the left or the right (not both). With that concept, we define what it means for a function to be continuous – a property of functions that is essential for calculus. Finally, we return to the idea of what it means for a function to be differentiable and connect this to the idea of continuity.

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).

• Evaluate a limit from the left and a limit from the right and use one-sided limits to tell if a (two-sided) limit exists.
• Define what it means for a function to be continuous at a point.
• Define what it means for a function to be differentiable at a point.
• Determine whether a function is continuous at a point by examining the graph of that function.
• Determine whether a function is differentiable at a point by examining the graph of that function.

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.

• Explain the relationship between continuity and differentiability. (If a function is continuous at a point, must it also be differentiable there? If a function is differentiable at a point, must it also be continuous there?)

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 1.7. 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: Limits, continuity, and differentiability (4:02)
• More limits (5:58)
• Determining continuity (7:22)
• Determining differentiability graphically (4:41)

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.