# The value of the numbers of a and b so that lim x → 0 a x + b − 2 x = 1 . ### Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805 ### Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805

#### Solutions

Chapter 2, Problem 2P
To determine

## To find: The value of the numbers of a and b so that limx→0ax+b−2x=1.

Expert Solution

The values of a=4 and b=4.

### Explanation of Solution

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.

Substitute b=4 in equation (1),

limx0ax+b2x=limx0ax+(4)4x(ax+(4)+2)=limx0axx(ax+4+2)=limx0a(ax+4+2)=a(a(0)+4+2)[by direction substitution]

Simplify further,

limx0ax+b2x=a4+2=a2+2=a4

The values of a and b satisfies the condition limx0ax+b2x=1.

limx0ax+b2x=a41=a4a=4

Thus, the value of the number a is 4.

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