Answered: In Problems 55-56, determine whether f is differentiable at by considering

Name: In Problems 55-56, determine whether f is differentiable at by conside
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Description: In Problems 55-56, determine whether f is differentiable at by considering

Calculus: Early Transcendentals

8th Edition

ISBN: 9781285741550

Author: James Stewart

Publisher: Cengage Learning

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In Problems 55-56, determine whether f is differentiable at by considering

Expert Solution

Step 1

(55)

$\begin{array}{rcl} \lim_{h \to 0^{+}} \frac{f (0 + h) - f (0)}{h} & = & \lim_{h \to 0^{+}} \frac{f (h) - f (0)}{h} \\ = & \lim_{h \to 0^{+}} \frac{|h| - |0|}{h} \\ = & \lim_{h \to 0^{+}} \frac{h - 0}{h} \\ = & \lim_{h \to 0^{+}} \frac{h}{h} \\ = & \lim_{h \to 0^{+}} 1 \\ = & 1 \end{array}$

Step 2

$\begin{array}{rcl} \lim_{h \to 0^{-}} \frac{f (0 + h) - f (0)}{h} & = & \lim_{h \to 0^{-}} \frac{f (h) - f (0)}{h} \\ = & \lim_{h \to 0^{-}} \frac{|h| - |0|}{h} \\ = & \lim_{h \to 0^{-}} \frac{- h - 0}{h} \\ = & \lim_{h \to 0^{-}} \frac{- h}{h} \\ = & \lim_{h \to 0^{-}} - 1 \\ = & - 1 \end{array}$

As $\begin{array}{rcl} \lim_{h \to 0^{-}} \end{array} \begin{array}{rcl} \end{array} \begin{array}{rcl} \frac{f (0 + h) - f (0)}{h} \end{array} \begin{array}{rcl} \neq \end{array} \begin{array}{rcl} \lim_{h \to 0^{+}} \end{array} \begin{array}{rcl} \end{array} \begin{array}{rcl} \frac{f (0 + h) - f (0)}{h} \end{array}$ , the limit does not exist

and hence the function is not differentiable at x=0.

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