   Chapter 4.4, Problem 92E ### 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 ) = { | x | x if   x ≠ 0 1 if   x = 0 (a) Show that f is continuous at 0.(b) Investigate graphically whether f is differentiable at 0 by zooming in several limes toward the point (0, 1) on the graph of f.(c) Show that f is not differentiable at 0. How can you reconcile this fact with the appearance of the graphs in part (b)?

(a)

To determine

To show: The function f is continuous at 0 if f(x)={|x|xif x01if x=0 .

Explanation

Proof:

To show that f is continuous at 0 it is sufficient to show that limx0f(x)=f(0) that is limx0f(x)=1 .

Let y=limx0|x|x . (1)

Take natural logarithm on both sides,

lny=ln(y=limx0|x|x)=limx0(ln|x|x)=limx0(xln|x|)=limx0ln|x|1x

Therefore, lny=limx0ln|x|1x . (2)

Obtain the value of the function as x approaches 0 .

As x approaches 0 , the numerator is,

ln|x|=ln|0|=

As x approaches 0 , the denominator is, 1x=

(b)

To determine

To check: Whether the function f(x)={|x|xif x01if x=0 is differentiable at 0 by zooming the graph towards the point (0,1) .

(c)

To determine

To show: The function f(x)={|x|xif x01if x=0 is not differentiable at 0; explain how it contradicts with the graphs appeared in the part (b).

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