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Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343

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BuyFindarrow_forward

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 3 = 0

To determine

To prove: The limit of a function limx0(x3) 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, 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, f(x)=x3, 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 |x30|<ε.

Therefore, find a number δ such that if 0<|x0|<δ then |x3|<ε if and only if |x|3<ε.

That is, if 0<|x0|<δ then |x|<ε3

So, choose a number δ=ε3

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