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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 and illustrate with a diagram like Figure 9.

FIGURE9

Chapter 2.4, Problem 15E, Prove the statement using the ,  definition of a limit and illustrate with a diagram like Figure 9.

lim x 3 ( 1 + 1 3 x ) = 2

To determine

To prove: The limit of a function limx3(1+13x) is equal to 2.

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=3, L=2 and f(x)=1+13x.

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

if 0<|x3|<δ then |(1+13x)2|<ε.

Consider |(1+13x)3|.

|(1+13x)2|=|13x1|=|x33|=|13||x3|=13|x3|

Therefore, to find a number δ such that if 0<|x3|<δ then 13|x3|<ε.

That is, if 0<|x3|<δ, then |x3|<3ε

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