Guided Practice 1.2: The Notion of Limit

Overview

In this section we introduce the all-important concept of the limit. A limit is a process that we apply to an existing function where we let the input to the function get closer and closer to – but not become exactly equal to – a certain input and observe what the outputs do. This process is central for computing instantaneous velocities, and it will lead us to the main concepts of calculus. For now, we will define the concept of the limit and calculate limits using graphical and numerical methods.

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).

• State the definition of what it means for a function $f$ to have a limit $L$ as $x$ approaches a number $a$, and explain what the definition means in nontechnical terms.
• Explain the meaning of the notation: $\displaystyle{\lim_{x \rightarrow a} f(x) = L}$ in clear terms.
• Calculate limits of functions (or determine if a function fails to have a limit) by examining a graph of the function.
• Calculate limits of functions (or determine if a function fails to have a limit) by constructing a table of values for the function (see Example 1.2).
• Explain the difference between the value of a function at a point, and the value of a limit of a function near a point.

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.

• Evaluate a limit of an expression that involves fractions algebraically, by simplifying the fraction and “taking limits”. (See Activity 1.4.)
• Apply the concept of a limit to find the instantaneous velocity of a moving object, by evaluating a limit of an average velocity.

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 1.2. Make sure to read actively, working through examples and activities as you go.

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

• Limits (6:02)
• Limits of functions using graphing tools (6:27)
• Limits of functions using tables (5:58)
• Limits of functions using spreadsheets (5:38)

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.