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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 2 x 3 = 8

To determine

To prove: The limit of a function limx2(x3) is equal to 8 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, a=2, L=8 and f(x)=x3.

By the definition of ε and δ, it is enough to find a number δ such that if 0<|x2|<δ then |x38|<ε.

Consider |x38|.

|x38|=|x3(2)3|=|(x2)(x2+2x+4)||x2||x2+2x+4|

If, there exists a positive constant C, such that |x2+2x+4|<C, then |x2||x2+2x+4|<C|x2|<ε.

Thus, find a number δ such that 0<|x2|<δ then |x2|<εC.

So, choose δ=εC.

If x lies in any interval centered at 2, the value C is obtained as follows.

Assume |x2|<1.

That is, 1<x2<1

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