   Chapter 2.1, Problem 59E

Chapter
Section
Textbook Problem
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# Determine whether f ′ ( 0 ) exists. f ( x ) = { x sin 1 x  if  x ≠ 0 0           if  x = 0

To determine

whether f'(0) exists

Explanation

Definition of derivative:

f'a=limh0f(a+h)-f(a)h

Calculations:

f'0 = limh0f(0+h)-f(0)h

= limh0hsin1h-0h

=limh0sin1h

The value of sin1h rapidly oscillates between takes -1 and 1 as h approaches 0

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