   Chapter 4.4, Problem 91E ### Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343

#### Solutions

Chapter
Section ### Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343
Textbook Problem

# Let f ( x ) = { e − 1 / x 2 if   x ≠ 0 0 if   x = 0 (a) Use the definition of derivative to compute f′(0).(b) Show that f has derivatives of all orders that are defined on ℝ . [Hint: First show by induction that there is a polynomial pn(x) and a nonnegative integer kn such that f ( n ) ( x ) = p n ( x ) f ( x ) / x k n for x ≠ 0.)

(a)

To determine

To compute: The value of f(0) by using definition of derivative.

Explanation

Given:

The function f is defined as, f={e1x2if x00if x=0 .

Calculation:

Use definition of derivative and observe that,

f(0)=limx0f(x)f(0)x0=limx0f(x)x

Hence, f(0)=limx0f(x)x . (1)

It is enough to show that limx0f(x)xn=0 for all integer n0 .

Let y=1x2 .

Consider the limit as follow.

limx0f(x)x2n=limx0e1x2(x2)n=limx01e1x2(x2)n=limyyney

Obtain the value of the function as y approaches .

As y approaches , the numerator is,

yn=n=

And the denominator is,

ey=e=

Thus, limyyney= is in an indeterminate form.

Therefore, apply L’Hospital’s Rule and obtain the limit as shown below

(b)

To determine

To show: On a set of real numbers derivatives of f for all orders are well defined.

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