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Which of the following function is differentiable at $x=0$?

$\begin{array}{1 1}(a)\;\cos(|x|)+|x|&(b)\;\cos(|x|)-|x|\\(c)\;\sin(|x|)+|x|&(d)\;\sin(|x|)-|x|\end{array}$

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1 Answer

Let $f(x)=\sin(|x|)-|x|$
$Lf'(0)=\lim\limits_{h\to 0}\large\frac{f(0-h)-f(0)}{-h}$
$\qquad\;\;=\lim\limits_{h\to 0}\large\frac{\sin h-h-0}{-h}\qquad$$ h > 0$
$\qquad\;\;=\lim\limits_{h\to 0}\large\frac{\sin h}{-h}$$+1=-1+1=0 $
Hence (d) is the correct answer.
answered Jan 2, 2014 by sreemathi.v
 

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