Evaluating Limits Analytically (Lesson 8.2 Day 3)
Unit 8 - Day 5
Use known limits to reason about unknown limits
Choose an appropriate strategy for evaluating a limit
Quick Lesson Plan
We just can’t get enough of these analytical limits! Evaluating limits from equations is notoriously hard for students, so we are going nice and slow, adding in 1-2 new strategies every day. Today students will learn about two new graphs and use additional algebraic techniques to evaluate limits.
Today’s handout has three different rows of limits, with the left-most column showing the most basic form of the limit and then getting progressively harder as they move across the row. Students start in their regular groups of 3-4 students. Have students number off in their groups by threes. Randomly assign the fourth group member to be either a 1, 2, or 3. We don’t want a ton of extra 1’s! Once students have a number, designate an area in the room for that group number to meet, preferably by a whiteboard. This new group of experts will now focus on only one of the three new types of limits. Group 1 does the first limit problem in row 1, group 2 does the first limit problem in row 2, and group 3 does the first limit problem in row 3. Encourage students to use any strategies they can, including making tables of values and looking at graphs. Tell students they should use their 8 minutes to explore how to evaluate the limit and then be confident enough in the method to teach someone else. They will soon get the chance to do just that!
After the eight minutes (give or take) disperse the expert groups back to their original groups (jigsaw groups). Every group member will now get a chance to share what they learned in their expert group and teach their group members how to do the first limit problem of their given row. Group members should be able to ask questions and the “expert” should be ready to answer them. Once everyone has done their mini-presentation, groups can build on what they learned by continuing on with limits in subsequent columns. These get slightly harder as they move further to the right. Students should see that knowing the original limit can help evaluate other limits, especially when you are able to identify the transformations that occurred to the parent function. In row 3, students will continue to use the strategy of expanding, then canceling, to evaluate the difference quotient (though they don’t know it’s called that yet!)
When students finish evaluating the limits, they should complete the reflection questions on the back to help them notice similarities in the limits in each row and in the strategies used for various types of limits.
Check in on groups to see how much needs to be added in the whole class debrief. If students understand how these limits work, a whole class discussion may not be needed. However, we do strongly suggest putting the two new parent functions of (sin x)/x and |x|/x on poster paper and hanging somewhere in the room. These limits will show up MANY more times this chapter, and students will need to know what these graphs look like.
Although we are not using formal notation around difference quotients and derivatives in this lesson, the limits in row 3 will help students prepare for the limit definition of the derivative in Unit 9. We decided to incorporate the procedural aspect into this lesson to demonstrate yet another method for evaluating limits analytically. We will build the concept of a derivative in great depth in Unit 9.