Module 9: LIMITS--EPSILON AND DELTA
Exploring Calculus

Introduction

In this discovery section we investigate limits more closely. We will see how close x has to be to a certain value a (called the -band about a) when we want f (x) to be within units of some y-value L (called the -band about L). See the graph below.

Critical Thinking Questions

1. Given the function.

a. Graph the function, using a full sheet of graph paper . Draw and label the epsilon - band and resulting delta - band using the particular values x = 1, = 1, and chosen according to your calculations.

= ____

    b.     Display the graph to help with the problems below and choose the same function, x-value, and as in Q.1a.

Find the largest which will work, accurate to the hundredths place. Does the result confirm your calculations in Q.1a ?

= ____

    c. For the same function and , does change if you change the x-value?

2. Given the function f (x) = 3x - 2

    a. Let x = 1 and = 0.1. Find the largestwhich will work.

=  ____

    b. For the same function and, does change if the x-value changes?

    c. What relationship between and can you discover from the last two examples?

3. Given the function y = x²

    a. Graph the function on each of the two grids below and let = 1. Draw the - and the resulting -bands for x = 1 on the first graph and for x = 2 on the second graph.

    b. What is the largest delta which will work in each case?

        x = 1     = ____

        x = 2     = ____

    c. Are your answers above confirmed by the computer's answers?

    d. Does your answer depend on the x-value chosen?

4. Now consider the function y = x³ with = 0.005.

Do not enter the function into the computer.

    a. What will happen to the largest value ofwhich will work as x increases from 1 to 3? Circle one of the choices below.

        1. It will be the same with each x-value.

        2. It will increase as x increases.

3. It will be the same for two x-values and be larger for the third.

4. It will be the same for two x-values and be smaller for the third.

    b. Repeat Q.4a on the computer. Do your answers correspond?

5. a. For which class of functions does the relationship between and stay the same for all x-values chosen?

    b. Why does the relationship between and stay the same for this type of function?

c. For all other classes of functions, the relationship between and changes as the x-value changes. Why do you think this happens?

Exploring Calculus