   Chapter 2.4, Problem 14E

Chapter
Section
Textbook Problem

Given that lim x → 2 ( 5 x − 7 ) = 3 , illustrate Definition 2 by finding values of δ that correspond to ε. = 0.1, ε = 0.05, and ε = 0.01.Definition 2 To determine

To find: The number of values of δ that correspond to ε=0.1,ε=0.05 and ε=0.01.

Explanation

Given:

The limit of the function limx2(5x7) is equal to 3.

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”.

Calculation:

Let ε be a given positive integer. Here, f(x)=5x7, a=2 and L=5.

Consider |(5x7)8|.

|(5x7)3|=|5x73|=|5x10|=|5||x2|=5|x2|

By the definition of ε and δ, it is enough to find a number δ such that,

if 0<|x2|<δ, then |(5x7)8|<ε

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