   Chapter 2.4, Problem 31E ### 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 → − 2 ( x 2 − 1 ) = 3

To determine

To prove: The limit of a function limx2(x21) is equal to 3 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=3 and f(x)=x21.

By the definition of ε and δ, it is enough to find a number δ such that if 0<|x(2)|<δ then |(x21)3|<ε.

Consider |(x21)3|

|(x21)3|=|x213|=|x24|=|x222|=|x+2||x2|

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

Thus, find a number δ such that 0<|x(2)|<δ then |x+2|<εC.

So, choose δ=εC.

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

Assume |x(2)|<1

1<x+2<15<x2<3

This implies that, |x2|<5

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