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* *Reflection: Complex Tasks
Techniques for Finding Limits - Section 2: Share and Explore

While students were working on finding the limit for one of the problems, a student asked a great question - is 0/0 the only indeterminate form or would something like 1/0 also be considered indeterminate form? This was something I definitely wanted to address with the entire class so I wrote it on the board so we could come back to it later.

During the discussion we thought of a really easy function that we could use to address this question (y = 1/x) and we saw that there was no limit when it was in this form. I like this process of jotting down the question on the board as it keeps a record of the important questions we need to address as a class.

# Techniques for Finding Limits

Lesson 2 of 13

## Objective: SWBAT algebraically find the limit of a function.

*50 minutes*

#### Launch

*15 min*

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.

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#### Share and Explore

*20 min*

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.

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#### Summarize

*15 min*

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.

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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 Limit of a Function
- LESSON 2: Techniques for Finding Limits
- LESSON 3: The Tangent Line Problem - Day 1 of 2
- LESSON 4: The Tangent Line Problem - Day 2 of 2
- LESSON 5: The Power Rule
- LESSON 6: Formative Assessment: Limits and Derivatives
- LESSON 7: Derivatives and Graphs
- LESSON 8: The Second Derivative
- LESSON 9: Maximizing Volume - Revisited
- LESSON 10: The Rock Problem
- LESSON 11: Unit Review: Limits and Derivatives
- LESSON 12: Unit Review Game: The Row Game
- LESSON 13: Unit Assessment: Limits and Derivatives