   Chapter 2, Problem 4RCC ### 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

# What does the Squeeze Theorem say?

To determine

To Describe: The Squeeze Theorem

Explanation

Statement:

Suppose that g(x)f(x)h(x)  for all x in some open interval containing c except possibly at c itself. If limxcg(x)=L=limxch(x) then limxcf(x)=L.

Result used: Definition of limits

Let ε>0 for any δ>0 such that |f(x)L|<ε whenever 0<|xc|<δ.

By the definition of limits, since limxcg(x)=L there exist some δ1>0 such that

|g(x)L|<ε for all 0<|xc|<δ1

Proof:

Let ε>0 for any δ>0 such that |f(x)L|<ε whenever 0<|xc|<δ.

By the definition of limits, since limxcg(x)=L there exist some δ1>0 such that

|g(x)L|<ε for all 0<|xc|<δ1.

Thus, ε<g(x)L<ε for all 0<|xc|<δ1

Lε<L+g(x)L<L+ε for all 0<|xc|<δ1

Lε<g(x)<L+ε for all 0<|xc|<δ1 (1)

Similarly, since limxch(x)=L there exist some δ2>0 such that

|h(x)L|<ε for all 0<|xc|<δ2.

Thus, ε<h(x)L<ε for all 0<|xc|<δ2

Lε<L+h(x)L<L+ε for all 0<|xc|<δ2

Lε<h(x)<L+ε for all 0<|xc|<δ2 (2)

Since g(x)f(x)h

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Study Guide for Stewart's Single Variable Calculus: Early Transcendentals, 8th 