# The value of lim x → 0 | 2 x − 1 | − | 2 x + 1 | x .

### 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 3P
To determine

## To evaluate: The value of limx→0|2x−1|−|2x+1|x.

Expert Solution

The limit of the function is –4.

### Explanation of Solution

Definition used:

An absolute function |x| is defined as follows.

|x|={x if x<0x if x0.

Calculation:

Obtain the value of the limit of the function limx0|2x1||2x+1|x.

Consider the function f(x)=|2x1||2x+1|x.

From the definition of absolute function,

|2x1|={(2x1)  if  2x1<0    2x1    if  2x10 and |2x+1|={(2x+1)     if  2x+1<0(2x+1)  if  2x+10

This follows that,

|2x1|={(2x1)  if  x<12    2x1    if  x12 and |2x+1|={(2x+1)  if  x<12   2x+1     if  x12

This implies that, |2x1||2x+1| is defined in the interval 12<x<12 as x approaches zero.

When x approaches 0, then |2x1| is negative. That is, |2x1|=(2x1). for x<12.

Similarly, when x approaches 0, then |2x+1| is positive. That is, |2x+1|=2x+1. for x>12.

Thus, the function f(x) is defined f(x)=(2x1)(2x+1)x as x approaches zero.

Take the limit of the function as x approaches zero.

limx0f(x)=limx0((2x1)(2x+1)x)=limx0((2x+1)(2x+1)x)=limx0(2x+12x1x)=limx0(4xx)

Simplifying further,

limx0|2x1||2x+1|x=4limx0(1)=4(1)=4

Thus the value of the limit of the function is 4.

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