What is a limit? (Lesson 8.1 Day 1)
Unit 8 - Day 1
Understand limits as predicted or intended outputs based on surrounding behavior
Evaluate limits using graphs and tables
Use limit notation to describe function behavior to the left and right of a particular x-value
Quick Lesson Plan
Today we set the scene for the big idea of limits. Students learn that limits are predictions or intentions for what the value of a function will be, based on surrounding data. The data students look at is the price of a gallon of gas. In question 2, the students are given a graph of gas price trends for Grand Rapids. The data for March 11th is intentionally missing as students will learn how to use the surrounding gas prices to make a prediction about what happens directly on March 11th. The big idea in question 2 is that it is impossible to make a reasonable prediction because the data from before March 11th does not coincide with the data from after March 11th. While some students may choose a price of gas in the middle, or attempt to draw in the missing piece of the graph, ask students how certain they are of their answer. Some students may even notice that the missing date was right around the beginning of our COVID shut-down, which further reinforces the idea of unpredictability.
In question 3, students are given a table of data values instead of a graph. Again, data for March 11th is missing. This time, there seems to be a consistent trend in gas prices leading students to the conclusion that the price of gas was $2.43. The data from before March 11th and the data from March 11th supports this conclusion.
In part b, we preview an upcoming idea of continuity by asking if the predicted value matches the actual value. It is critical that students understand that their prediction is still valid based on the data, even if the price of gas was much higher than anticipated. The limit still exists! Many lessons in this chapter will build on this idea as we discuss how to evaluate limits at holes and classify these kinds of discontinuities as removable.
The formalization of this lesson introduces formal limit notation. Be sure to say the limit statements out loud multiple times using the correct language: “The limit as t approaches 11 from the left of G(t) is ___”
Continue to reinforce that a limit isn’t asking for the actual output, a limit asks for the intended output. In our class we talk about the old adage “It’s the thought that counts”. Even when a plan goes awry, we can think about the person’s intention and what they were going for. This is exactly the idea of a limit.
In the Important Ideas section we provide the formal definition of what it means for a limit to exist. Explain to students why the two statements are biconditional; why does one imply the other? Also make note that we consider the value of the limit L to be finite.
For students struggling to evaluate limits from graphs, have them use their finger to “jump on the curve” somewhere to the left of the desired x-value and follow the curve all the way until the given x-value, and see what y-value is being approached. Repeat this from the right side. We will spend much more time practicing evaluating limits from graphs in tomorrow’s Quiz, Quiz, Trade activity.