# Whether f ′ ( 0 ) exits or not.

### 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 2.6, Problem 53E
To determine

## Whether f′(0) exits or not.

Expert Solution

The limit does not exist.

### Explanation of Solution

Definition:

A function f(x) is differentiable at point x, then f(x)=limh0f(x+h)f(x)h.

Calculation:

Consider the function f(x)=xsin(1x) when x0 and f(0)=0,

By definition, f(x)=limh0f(x+h)f(x)h

Substitute x=0 in the above equation,

f(0)=limh0f(0+h)f(0)h=limh0f(h)f(0)h=limh0hsin(1h)0h=limh0sin(1h)

Since sin(1h) is values between 1 to 1 in any interval containing 0.

Therefore, the limit does not exist.

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