   Chapter 2, Problem 26RE

Chapter
Section
Textbook Problem

Prove the statement using the precise definition of a limit. lim x → 0 x 3 = 0

To determine

To prove: The limit of a function limx0(x3) is equal to 0.

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”.

To guess: The number δ.

Let ε be a given positive integer. Here, a=0, L=0 and f(x)=x3.

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

if 0<|x0|<δ then |(x3)0|<ε.

Consider, |(x3)0|<ε.

|(x3)|<ε

Take cubes on both sides of the inequality,

|(x3)|3<ε3|(x3)3|<ε3|x|<ε3

Therefore, to find a number δ such that if 0<|x|<δ then |x|<ε3

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