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Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343

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Chapter
Section
BuyFindarrow_forward

Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343
Textbook Problem

Use a graph to find a number δ such that

if | x π 4 | < δ then    | 2 x x 2 + 4 0.4 | < 0.1

To determine

To find: The number δ by using the graph.

Explanation

Given:

To find a number δ such that if |x1|<δ then |2xx2+40.4|<0.1.

Definition used:

“Let f be a function defined on some open interval that contains the number a, except possibly at a itself. Then, the limit of f(x) as x approaches a is L, limxaf(x)=L if for every number ε>0 there is a number δ>0 such that if 0<|xa|<δ then |f(x)L|<ε”.

Estimation:

Obtain the number δ such that if |x1|<δ then |2xx2+40.4|<0.1 by using the graph.

By the definition of limit, the values of a=1, L=0.4, f(x)=2xx2+4 and ε=0.1

Rewrite the inequality |2xx2+40.4|<0.1 as follows.

0.1<2xx2+40.4<0.10.1+0.4<2xx2+40.4+0.4<0.1+0.40.3<2xx2+4<0.5

The values of x for the function y=2xx2+4 lies between the horizontal lines y=0.3 and y=0.5.

Use the online graphing calculator to draw the function y=2xx2+4, y=0.3 and y=0.5 near the point (1,0

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