   Chapter 2, Problem 2P ### 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

# Find numbers a and b such that lim x → 0 a x + b − 2 x = 1.

To determine

To find: The value of the numbers of a and b so that limx0ax+b2x=1.

Explanation

Calculation:

Obtain the value of the numbers of a and b.

Consider the function f(x)=ax+b2x.

Rationalize the numerator,

f(x)=ax+b2x×ax+b+2ax+b+2=(ax+b)2(2)2x(ax+b+2)=ax+b4x(ax+b+2)

Take the limit of the function f(x) as x approaches 0.

limx0f(x)=limx0ax+b4x(ax+b+2) (1)

Here, the denominator approaches 0 as x approaches 0, so the limit exist only if the numerator of the function must be equal to zero as x approaches 0. That is,

limx0(ax+b4)=0a(0)+b4=0b4=0b=4

Thus, the value of the number b=4

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