The purpose of today's lesson is to formalize students' thinking about limits and the value that a sequence approaches. Limits have important implications in calculus, so this is a good time to introduce students to vocabulary and notation associated with the concept.
To begin the lesson, I ask students to think about questions #1 and #2 on the worksheet with their table. I give them 3-4 minutes to consider these questions and I will elicit responses after that time.
Students have a lot of intuitive sense about limits. It is not surprising that the sequence listed in question #1 is approaching zero, so when I ask students for the answer that is usually their first response. As I was walking around I heard one student say to another student: "The limit is going to be zero, just like in the gummy bear problem." It doesn't get any better than this - hearing a student bring up a past context to make sense of a new, abstract problem!
Then I ask if the sequence approaches -1. I'll play devil's advocate and they will agree with me that each term is getting closer and closer to -1. When I ask them why 0 is the limit and not -1, it is difficult for students to put it into words. This frames the lesson and presents the need for us to find a mathematical definition for a limit.
Next, I tell my students that we are going to be working on finding a formal definition for a limit. To do this, we are going to be using open intervals centered at certain values to investigate the concept of a limit. I review what an open interval is and have students work on question #3 from the worksheet with their table groups. If students are having difficulty thinking about how many terms are outside of the interval, I tell them to sketch a number line so they have a visual representation of the interval.
After students have sufficient time to work on #3, we will go through the answers together. Part d is challenging, so they may need a little nudge to come to the conclusion that there will always be a finite number of terms outside of the open interval when it is centered at the limit.
In the video below, I discuss how I move from the investigation with intervals to a formal definition of a limit.
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Here is a link with some information about the limit properties that I add to the worksheet with students.
After the lesson, I will explain to students that limits are an important concept in calculus and that they will help us solve problems in the near future. They may seem simple and unimportant to students right now, but I hope they will understand the power of a limit once we look at some applications in later lessons.
For homework, students will complete questions #1-14 from the Notes - Limits worksheet.