We studied limits of sequences earlier in the year, but today we are going to extend that knowledge and think about limits of functions. The skill set is definitely similar, but adding a graphical approach will enhance student understanding.
I start the lesson by activating the prior knowledge of a limit of a sequence by having students work in their table groups on questions #1 from this worksheet. We will share our responses after students get their answers.
At this point I want to transition to the limit of a function, so I will explain to students that we are going to be finding the limits of graphs and equations instead of sequences. Next I will define the limit of a function and introduce the notation (including notation for right side and left side limits). I make a big deal about the fact that the limit must approach the same value from both sides in order to exist.
After the initial phase of defining the limit, I try not to give too much instruction and just have students go at questions #2 and #3 on the worksheet. They are using graphs and equations to find limits, but I do not give them any method to find the limit - I want to see where their intuition takes them.
As students are working I will ask them questions to get a feel for the reasons that they are giving for the limits they choose. I am also making mental notes about the techniques that are most prevalent. Below are a list of common misconceptions that I see when working on this lesson. Hopefully you can use this list to anticipate the misconceptions and be able to guide them to better thinking.
When it is time to share the answers for #2 on the worksheet, I will have students share their answers with the class and give their reasons. I will usually randomly select students and hope that a few of them got the wrong answers - the discussions about the validity of wrong answers are usually very beneficial to the class. I talk more about this in the video below.
For questions #3, I make sure to get a variety of different methods by selecting the students to share their thinking. Here are some different methods I try to elicit:
Next, for question #4 I will go over the nuts and bolts of using a table on the graphing calculator to evaluate the limit. Many students will need assistance setting up the table and changing the setting so that they can enter in the x-values instead of it being automatic.
To close the lesson, I ask students about the similarities and differences in finding the limit of a sequence and the limit of a function. I want to reinforce the idea that in both cases we are looking at what value is being approached, but that the representations are different. Finally, I will assign some limit problems from their textbook as homework.