## Notes - Limits.docx - Section 2: Explore

# The Limit of a Sequence

Lesson 12 of 18

## Objective: SWBAT find the limit of a sequence.

*50 minutes*

#### Launch

*10 min*

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.

#### Resources

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#### Explore

*25 min*

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**.

Here is a link with some information about the limit properties that I add to the worksheet with students.

#### Resources

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#### Summarize and Extend

*15 min*

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.

#### Resources

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- UNIT 1: Functioning with Functions
- UNIT 2: Polynomial and Rational Functions
- UNIT 3: Exponential and Logarithmic Functions
- UNIT 4: Trigonometric Functions
- UNIT 5: Trigonometric Relationships
- UNIT 6: Additional Trigonometry Topics
- UNIT 7: Midterm Review and Exam
- UNIT 8: Matrices and Systems
- UNIT 9: Sequences and Series
- UNIT 10: Conic Sections
- UNIT 11: Parametric Equations and Polar Coordinates
- UNIT 12: Math in 3D
- UNIT 13: Limits and Derivatives

- LESSON 1: The Skyscraper Problem
- LESSON 2: The Fractal Tree
- LESSON 3: Describing Sequences and Series
- LESSON 4: Arithmetic Sequences
- LESSON 5: Geometric Sequences
- LESSON 6: The Fractal Tree Revisited
- LESSON 7: Investments, Loans, and Mortgages - Day 1 of 2
- LESSON 8: Investments, Loans, and Mortgages - Day 2 of 2
- LESSON 9: Mathematical Induction
- LESSON 10: Formative Assessment Review: Sequences and Series
- LESSON 11: Formative Assessment: Sequences and Series
- LESSON 12: The Limit of a Sequence
- LESSON 13: Area Under a Curve - Day 1 of 2
- LESSON 14: Area Under a Curve - Day 2 of 2
- LESSON 15: Binomial Expansion
- LESSON 16: Unit Review: Sequences and Series
- LESSON 17: Unit Review Game: Pictionary
- LESSON 18: Unit Assessment: Sequences and Series