   Chapter 2.4, Problem 34E

Chapter
Section
Textbook Problem

Verify, by a geometric argument, that the largest possible choice of δ for showing that lim x → 3 x 2 = 9 is δ = 9 + ∈ − 3

To determine

To show: The value of δ=9+ε3.

Explanation

Definition used:

The definition of epsilon-delta of the limit:

Let f be a real-valued function defined on a subset D of the real numbers. Let c be the limit point of D and let L be a real number limxcf(x)=L.

If for every ε>0 there exist a δ such that, for all xD, if 0<|xc|<δ, then |f(x)L|<ε.

Formula used:

Difference of the two squaring variables a2b2=(a+b)(ab)

Graph:

Calculation:

The given limit of the function limx3(x2)=9.

By the definition of epsilon-delta of the limit:

Let ε>0 there exist δ>0 such that,

if 0<|x3|<δ, then |x29|<ε

if 0<|x3|<δ, then |(x+3)(x3)|<ε

if 0<|x3|<δ, then |x+3||x3|<ε

Let ε>0 that implies |x3|<δ

δ<x3<δ3δ<3+x3<3

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