   Chapter 2, Problem 25RE ### Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343

#### Solutions

Chapter
Section ### Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343
Textbook Problem

# Prove the statement using the precise definition of a limit. lim x → 2 ( 14 − 5 x ) = 4

To determine

To prove: The limit of a function limx2(145x) is equal to 4.

Explanation

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”.

Proof:

Let ε be a given positive integer. Here, a=2, L=4 and f(x)=145x.

By the definition of ε and δ, it is enough to find a number δ such that, if 0<|x2|<δ then |(145x)4|<ε.

Consider |(145x)4|<ε.

Simplify the expression |(145x)4| as follows.

|(145x)4|=|145x4|=|105x|=|5||x2|=5|x2|

Therefore, find a number δ such that if 0<|x2|<δ then 5|x2|<ε

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