# UTPA STEM/CBI Courses/Business Math/Business Calculus/Limits

Course Title: Limits with Applications to Business and Economics
Lecture Topic: Challenge-Based Instruction Faculty Development Workshop
Instructor: Jeffrey Castaneda
Institution: Texas A&M International University

## Backwards Design

Course Objectives
* Primary Objectives- By the next class period students will be able to:
** Discuss the behavior of values for a function f when x approaches close but not equal to a certain value.
** Evaluate/analyze limits with different sets of functions.
** Identify and apply properties of limits.
* Sub Objectives- The objectives will require that students be able to:
** Understand the theorems and definitions used in evaluating limits.
** Use techniques either algebraically or graphically to evaluate limits.
* Difficulties- Students may have difficulty:
** Knowing if a limit exist or does not exist especially for the indeterminate form case.
** Analyzing one-sided limits graphically.
* Real-World Contexts- There are many ways that students can use this material in the real-world, such as:
** Business and economics
** Life Sciences
** Social Sciences
Model of Knowledge
* Concept Map
** One-sided limits
** Two-sided limits
** Existence of limits
** Properties of limits
* Content Priorities
** Enduring Understanding
*** Evaluating limits with different types of functions and determining if the limits exist or does not exist.
*** Determining one-sided limits as well as two-sided limits.
*** Applying limits with real world problems.
** Important to Do and Know
*** The existence of a limit.
*** Using the properties of limits.
*** The indeterminate form.
*** The limit of a quotient.
*** The limits for polynomial and rational functions.
*** Construction of tables to guess the value for one-sided and/or two-sided limits.
** Worth Being Familiar with
*** The precise definition of a limit.
*** The theoretical concepts of a limit.
Assessment of Learning
* Formative Assessment
** In Class (groups)
The instructor will provide examples of Finding limits from the textbook and assign practice problems for students to understand how to evaluate limits.
Students will work the practice problems in groups of two
* Summative Assessment
** Assigned exams will help the students understand the applications of finding limits of a function.

## Legacy Cycle

OBJECTIVE
By the next class period, students will be able to:
* Discuss the behavior of values for a function f when x approaches close but not equal to a certain value.
* Evaluate/analyze limits with different sets of functions.
* Identify and apply properties of limits.
The objectives will require that students be able to:
* Understand the theorems and definitions used in evaluating limits.
* Use techniques either algebraically or graphically to evaluate limits.
THE CHALLENGE
Suppose a company manufactures sport watches. Each week the company produces about 5 sport watches. Furthermore, the company also presents the cost function for producing sport watches, say x given by C(x) = 1; 000 + 4x 0 _ x _ 100: Your task is to evaluate the limit for the cost function when the production level of sport watches approaches to 10.
GENERATE IDEAS
First, students will work the challenge problem individually and will need to ask themselves the following questions:
What are limits?
How to evaluate limits?
Does the limit exist?
Second, students will get in groups of two and present their solution to the challenge problem within the group.
MULTIPLE PERSPECTIVES
Students can also view videos from other professors in other institution discussing and explaining limits.
RESEARCH & REVISE
Students can conduct some research to solve the challenge problems. Activities include:
* Reading textbook.
* Reviewing instructor lecture notes.
* Searching webpages of lecture notes from other institution by other professors/instructors.
Students can revise the challenge problem and answer the problem algebraically by using the properties of limits graphically by graphing the function on paper or by calculator.
In addition, students may also use any software programs or mathematical language programs such as Excel or MATLAB to graph the function from the challenge problem.
Construct two tables to guess the value for one-sided limits. Solve other challenge problems either in class or homework. Take exams similar to the challenge problems and homework problems dealing with evaluating limits. Revisiting back to the challenge problem, we are finding the limit. i.e., lim x!10 1; 000 + 4x: Students can consider finding the one-sided limit, i.e., lim x!10+ 1; 000 + 4x and lim x!10􀀀 1; 000 + 4x:
Construct two tables x C(x) 10.5 1,042 10.10 1,040.4 10.08 1,040.32 10.01 1,040.04 10.001 1,040.004 x C(x) 9.5 1,038 9.80 1,039.2 9.90 1,039.6 9.99 1,039.96 9.999 1,039.996 Since lim x!10+ 1; 000 + 4x = 1; 040 and lim x!10􀀀 1; 000 + 4x = 1; 040; then lim x!10 1; 000 + 4x = 1; 040:
TEST YOUR METTLE
Students will be able to solve other challenge problems either in class or homework. They will also be able to take exams similar to the challenge problems and homework problems involving the evaluation of limits.
Revisiting back to the challenge problem, we are finding the limit,
i.e., lim x!10 1; 000 + 4x: Students can consider _nding the one-sided limit, i.e., lim x!10+ 1; 000 + 4x and lim x!10􀀀 1; 000 + 4x: Construct two tables x C(x) 10.5 1,042 10.10 1,040.4 10.08 1,040.32 10.01 1,040.04 10.001 1,040.004 x C(x) 9.5 1,038 9.80 1,039.2 9.90 1,039.6 9.99 1,039.96 9.999 1,039.996 Since lim x!10+ 1; 000 + 4x = 1; 040 and lim x!10􀀀 1; 000 + 4x = 1; 040; then lim x!10 1; 000 + 4x = 1; 040:
GO PUBLIC
One group will present the solution to the challenge problem on the board explaining to other students and instructor what they have researched and understand about limits. Later on, each group will be given different challenge problems and have to present their solutions in class.

## Pre-Lesson Quiz

1 Evaluate limx!4 x2 􀀀 16 x 􀀀 4 2 Does the limx!2 x 􀀀 1 x 􀀀 2 exist? 3 Evaluate limx!3 x2 􀀀 2x + 5 4 Does the limx!5 x 􀀀 5 x 􀀀 3 exist?

## Test Your Mettle Quiz

With the challenge problem above, evaluate the limit for the cost function when the number of sport watches approaches to 20.