Evaluating Limits Analytically (Lesson 8.2 Day 1)

Unit 8 - Day 3

​Learning Objectives​
  • Use direct substitution to evaluate limits

  • Evaluate limits of piecewise functions

  • Determine when direct substitution will work as a strategy for evaluating limits

Quick Lesson Plan
Activity: How Fast is Ayo Burning Calories?



Lesson Handout

Answer Key

Experience First

In this lesson students look at how fast Ayo is burning calories on a treadmill at various intervals during his exercise routine. Students are given an equation instead of a graph and first describe what the shape of the curve tells about his calorie burn. To figure out how fast he is burning calories near the 2 min mark, students fill out a table at t values near 2. Students see that the value being approached as t gets closer and closer to 2 is actually the same as just finding the output at t=2. This leads students to the first important analytical strategy for evaluating limits: direct substitution.


In questions 5-7 students think about Ayo’s calorie burn between t=4 and t=15, presented as a piecewise function. Students should realize that first Ayo is burning calories at a constant rate, and then this rate is decreasing as he is nearing the end of the workout. Ask students to make sense of this information in light of the workout. Make sure students understand that f(t) represents the rate of calorie burn, not the actual number of calories burned, as this would not stay constant. The key takeaway from this part of the activity is that there is a sudden jump in the curve at t=12. This means the limit does not exist even though f(12) exists. Drawing the piecewise graph in question 7 reinforces this idea. Be looking for students that try to draw a vertical or very steep line to get from the constant rate to the linear rate. Ask them if it is possible for him to be burning calories at two different rates at one instant in time. Students should realize the need to use a hole as a placeholder when beginning the linear graph.

Formalize Later

While this lesson already uses some limit notation that students are familiar with from previous lessons, the new learning is about how to evaluate that limit when just given an equation of a function. In yesterday’s card sort, most groups matched the limits to the graph and then the equations to the graphs. Moving forward, connecting all three of these representations, including going directly from equations to limits, will become even more important.


Ask students why direct substitution worked as a strategy for evaluating the limit at t=2 but not at t=12. Even though we are not using any formal ideas of continuity yet, students should be able to talk about smooth curves versus curves that jump. When there is no hole, gap, or asymptote in the graph, the actual y-value of the function and the intended y-value (the limit) are actually the same, so evaluating the function by direct substituting also gives the value of the limit! We use the language of a “well-behaved” function to talk about functions that act predictably and “play nice”. Tomorrow we will look at rational functions that do not behave quite so nicely.


You may notice that these lessons progress much more slowly than they would in AP Calculus. This is intentional! In Precalculus the goal is to build a conceptual framework around the big idea of limits and to explore at length the variety of strategies we have to evaluate limits and when each one might be appropriate. We continue to move back and forth between informal language and formal limit notation.