Which of the following is differentiable at x | vedclass

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(D) function $f(x)$ is differentiable at $x=0$ if $\lim_{h 	o 0^+} \frac{f(h)-f(0)}{h} = \lim_{h 	o 0^-} \frac{f(h)-f(0)}{h}$.For $f(x) = \sin |x| -

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$f(x)=\sin |x|-|x|$

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(D) function $f(x)$ is differentiable at $x=0$ if $\lim_{h 	o 0^+} \frac{f(h)-f(0)}{h} = \lim_{h 	o 0^-} \frac{f(h)-f(0)}{h}$.For $f(x) = \sin |x| - |x|$,we have $f(0) = \sin(0) - 0 = 0$.Right Hand Derivative $(RHD)$ at $x=0$:$\lim_{h 	o 0^+} \frac{\sin |h| - |h| - 0}{h} = \lim_{h 	o 0^+} \frac{\sin h - h}{h} = \lim_{h 	o 0^+} (\frac{\sin h}{h} - 1) = 1 - 1 = 0$.Left Hand Derivative $(LHD)$ at $x=0$:$\lim_{h 	o 0^-} \frac{\sin |-h| - |-h| - 0}{h} = \lim_{h 	o 0^-} \frac{\sin(-h) - (-h)}{h} = \lim_{h 	o 0^-} \frac{-\sin h + h}{h} = \lim_{h 	o 0^-} (-\frac{\sin h}{h} + 1) = -1 + 1 = 0$.Since $LHD = RHD = 0$,the function $f(x) = \sin |x| - |x|$ is differentiable at $x=0$.

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

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