   Chapter 1.6, Problem 38E

Chapter
Section
Textbook Problem

If 2x ≤ g(x) ≤ x4 − x2 + 2 for all x, evaluate lim x → 1 g ( x ) .

To determine

To evaluate: The limit value of the function g(x) as x approaches 1.

Explanation

Given:

The inequality is, 2xg(x)x4x2+2 for all x.

Limit Laws:

Suppose that c is a constant and the limits limxaf(x) and limxag(x) exist. Then

Limit law 1: limxa[f(x)+g(x)]=limxaf(x)+limxag(x)

Limit law 2: limxa[f(x)g(x)]=limxaf(x)limxag(x)

Limit law 3: limxa[cf(x)]=climxaf(x)

Limit law 7: limxac=c

Limit law 8: limxax=a

Limit law 9: limxaxn=an where n is a positive integer.

Theorem used: The Squeeze Theorem

“If f(x)g(x)h(x) when x is near a (except possibly at a) and limxaf(x)=limxah(x)=L then limxag(x)=L.”

Calculation:

Apply the Squeeze Theorem and obtain a function f smaller than g(x) and a function h bigger than g(x) such that both f(x) and h(x) approaches 4.

The given inequality becomes, 2xg(x)x4x2+2.

When the limit x approaches to 1, the inequality becomes,

limx12xlimx1g(x)limx1x4x2+2

Let f(x)=2x and h(x)=x4x2+2

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