Calculus (3rd Edition)

Published by W. H. Freeman
ISBN 10: 1464125260
ISBN 13: 978-1-46412-526-3

Chapter 3 - Differentiation - 3.2 The Derivative as a Function - Exercises - Page 115: 59


$f (x)$ is not differentiable at $x = 2$

Work Step by Step

Given $$f(x)=|x^2-4| $$ Since $$ f^{\prime}(x) =\lim _{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}$$ Then \begin{align*} f^{\prime}(2)&=\lim _{h \rightarrow 0} \frac{f(2+h)-f(2)}{h}\\ &=\lim _{h \rightarrow 0} \frac{\left|(2+h)^{2}-4\right|-0}{h}\\ &=\lim _{h \rightarrow 0} \frac{\left|h^{2}+4 h\right|}{h}\\ &=\lim _{h \rightarrow 0}|h+4| \cdot \frac{|h|}{h}\\ &= \lim _{h \rightarrow 0}|h+4| \lim _{h \rightarrow 0} \frac{|h|}{h}\\ &=4\lim _{h \rightarrow 0} \frac{|h|}{h} \end{align*} We consider the one-sided limits: $$\lim _{h \rightarrow 0^+} \frac{|h|}{h}=1,\ \ \ \ \lim _{h \rightarrow 0^-} \frac{|h|}{h}=-1 $$ Since the limits are not equal, then $f (x)$ is not differentiable at $x = 2$
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