Discuss the differentiability of $e^{-|x|}$ ?
I tried something like making $2$ domains, $x\ge 0$ and $x<0$
$e^{-|x|} = e^{-x} | x\ge 0 $
and
$e^{-|x|} = e^{x} | x<0 $
By using definition of differentiability,
$$\lim_{x \to 0^+}\frac{f(x)-f(0)}{x-0} = \lim_{x \to 0^+}\frac{e^{-x}-1}{x} = -1$$
and Similarly
$$\lim_{x \to 0^-}\frac{f(x)-f(0)}{x-0} = \lim_{x \to 0^-}\frac{e^{x}-1}{x} =+1$$
Hence, I can say that it is not differentiable at $x=0$.
Is my understanding right or Am I missing something ?
No comments:
Post a Comment