Defining Limits and Using Limit Notation (Topics 1.2-1.4)

Unit 1 - Day 2

​Learning Objectives​
  • Represent and interpret limits analytically using correct notation, including one-sided limits

  • Estimate limits of functions using graphs or tables

​Success Criteria
  • I can explain the idea of a limit.

  • I can use proper limit notation.

  • I can use tables and graphs to estimate limits.

Quick Lesson Plan
Activity: Can You Shoot Free Throws Like Nash? 
     

​Additional Materials

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Lesson Handout

Answer Key

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Overview

This lesson introduces the concept of a limit to students using the analogy of a frozen TV screen that causes us to miss some of the action. Luckily, by knowing what happens before and after we can piece together what likely happened to the basketball during that particular moment. Similarly, limits help us make accurate predictions for something we can’t easily measure. Analyzing tables and graphs of functions can give us the information we need about the behavior of the function at a certain point, even if the function is not actually defined at that point.

Teaching Tips

Make sure you cut out the data cards so they are scrambled before handing them out to students. The purpose of this portion of the activity is for students to realize that it would be beneficial to order the cards chronologically, getting closer and closer to 5 from the left, and then closer and closer to 5 from the right. By knowing the behavior of the ball very close to 5 milliseconds, we can predict the height of the ball when the video froze.

In our classroom, when a limit doesn’t exist, we require our students to write “DNE because…” and then justify their response. They must communicate how the function fails the definition (either because the left and the right limits don’t match, or because the function is unbounded at that point).

The last question on the Check Your Understanding portion is going to be important moving forward. Tell your students to memorize this!

Student Misconceptions

It is important for students to differentiate between a limit, which is an intended y-value, and the actual y-value at a particular x-value. Sometimes, the y-value is the same thing as the intended y-value, as students see in this lesson, but often times it is not. It’s important that students get repeated exposure to both cases. Often after students see a lot of graphs with holes, jumps, and asymptotes, they forget about the simpler case of well-defined, continuous functions.

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