   Chapter 2.4, Problem 40E

Chapter
Section
Textbook Problem

By comparing Definitions 2, 3, and 4, prove Theorem 2.3.1.Definition 2 Definition 3 Definition 4 To determine

To prove: The limit of the function limxaf(x)=L if and only if

limxaf(x)=L=limxa+f(x).

Explanation

Definition of a limit:

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 is,

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.

Definition of Left-Hand limit:

The Left-Hand limit of the function is limxaf(x)=L

if for every number ε>0 there is a number δ>0, such that

if aδ<x<a then |f(x)L|<ε.

Definition of Right-Hand limit:

The Right-Hand limit of the function is limxa+f(x)=L

if for every number ε>0 there is a number δ>0 such that

if a<x<a+δ then |f(x)L|<ε.

Proof:

Suppose that limxaf(x)=L then given ε>0 there exists δ>0 so that 0<|xa|<δ such that |f(x)L|<ε. Then aδ<x<a implies 0<|xa|<δ so |f(x)L|<ε.

Thus, limxaf(x)=L.

Also a<x<a+δ implies 0<|xa|<δ so |f(x)L|<ε

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