# Whether the statement “If f ( x ) &gt; 1 for all x and lim x → 0 f ( x ) exists, then lim x → 0 f ( x ) &gt; 1 ” 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 18RQ
To determine

## To find: Whether the statement “If f(x)>1 for all x and limx→0f(x) exists, then limx→0f(x)>1 ” is true or false.

Expert Solution

The statement “If f(x)>1 for all x and limx0f(x) exists, then limx0f(x)>1 ” is false.

### Explanation of Solution

Given information:

The given statement is “If f(x)>1 for all x and limx0f(x) exists, then limx0f(x)>1 ””.

Calculation:

Let us assume a function f(x)=x2+1 for all x(0,) .

So,

x>0x2>0x2+1=0

Further,

f(x)>1 , x(0,)

limx0f(x)=limx0f(x2+1)limx0f(x)=limx2+limx01limx0f(x)=1limx0f(x)=1

It is not necessary that limx0f(x)>1 , so the statement is false.

Therefore, the statement “If f(x)>1 for all x and limx0f(x) exists, then limx0f(x)>1 false.

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