   Chapter 2.4, Problem 25E ### 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 ε, δ definition of a limit. lim x → 0 x 2 = 0

To determine

To prove: The limit of a function limx0(x2) is equal to 0 by using the ε,δ definition of a limit.

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.

That is, 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, f(x)=x2, a=0 and L=0.

By the definition of ε and δ, it is enough to find a number δ such that if 0<|x0|<δ then |x20|<ε.

Therefore, find a number δ such that if 0<|x0|<δ, then |x2|<ε if and only if x2<ε

That is, if 0<|x0|<δ then |x|<ε

So, choose a number δ=ε

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