# Whether the statement, lim x → 4 ( 2 x x − 4 − 8 x − 4 ) = lim x → 4 ( 2 x x − 4 ) − lim x → 4 ( 8 x − 4 ) is true or false.

### 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 1RQ
To determine

## Whether the statement, limx→4(2xx−4−8x−4)=limx→4(2xx−4)−limx→4(8x−4) is true or false.

Expert Solution

The statement is false.

### Explanation of Solution

Difference law: Suppose that the limits limxaf(x) and limxag(x) exist.

Then limxa[f(x)g(x)]=limxaf(x)limxag(x).

Reason:

Let the function f(x)=2xx4 and g(x)=8x4.

Then, by the difference rule, the given statement is true only if the individual limits are exists.

That is, limx4(2xx48x4)=limx4(2xx4)limx4(8x4) is true whenever limx4(2xx4) and limx4(8x4) are exists.

The limit f(x)=2xx4 as x approaches 4 is computed as follows,

limx4(2xx4)=2(4)44=80

It is in indeterminate form. Thus, the limit does not exists.

The limit g(x)=8x4 as x approaches 4 is computed as follows,

limx4(8x4)=844=80

Since it is in an indeterminate form, the limit does not exist.

Since the individual limit does not exist, the difference law cannot be used here.

Therefore, the statement is false.

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