limits - Discuss the differentiability of $e^{-|x|}$?

Discuss the differentiability of $e^{-|x|}$ ?

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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 $x=0$.

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Is my understanding right or Am I missing something ?