Today's lesson is a continuation of our introduction to limits. Yesterday we used informal techniques to find limits (looking at a graph or table), and today we are going to use some algebraic techniques to evaluate.
I start class by giving students this task worksheet and having them work on just the front side with their table. The purpose of these questions are to review what we learned about limits and to refresh their memory about rational functions and discontinuities. I give students 10-15 minutes to work on the front side with their table groups.
Question #5 is really pushing students to algebraically evaluate the limit by factoring and cancelling out the common factor. Hopefully students will realize from questions #3 and #4 that canceling common factors is a viable option. If my students only use a graph or table to evaluate the limit, I will direct them to questions #3 and #4 to see if that will help them evaluate a limit.
Once students have had time to work on the front side, I want to discuss questions #4 and #5 of the worksheet together as a class. Here are a few things I want students to realize.
#4 - It is important that students realize that the graphs of f(x) and g(x) are identical except for the fact that g(x) has a removable discontinuity. It is important that students know this so they can still evaluate limits when a function is undefined at a point.
#5 - Here I am trying to bridge to the algebraic technique of finding a limit. I still want to talk about using the graph and the table to find the limit, but the algebraic method will take center stage. Usually I will choose a student to explain their thinking about it to the class. After the student presents their work, I will introduce the vocabulary term indeterminate form and discuss its meaning. More on this in the video below.
Next, we will transition to #6. Again, I want to focus on the algebraic method, so I instruct students that they cannot use a table or graph to find the limit. Teacher note: sometimes it helps to have them put their graphing calculator away while working on this. I will usually give students about 7 minutes to work on this with their table groups.
Once students realize that they cannot easily factor, they try to look for a different method. Usually a handful of students will think of multiplying the numerator and denominator by the conjugate. If many students are stuck I will have the students who figured it out share their method with others. After that time, we will share our work as a class.
Question #7 of the worksheet is a good summary of different ways to find limits. I will give students about 10 minutes to work on these. When we share our answers, I want to focus on the method students used - did they use a table, graph, or algebraic method?
Finally, students recap the algebraic techniques we used. I want students to walk away knowing that factoring and multiplying by the conjugate are two methods they can use if the limit is undefined at a point.