How do I check the differentiability of a $f(x) = x|x|$ at $x_0=0$?
I used the definition of a derivative and came up with \begin{align} f^\prime &= \lim_{x\to0}\dfrac{f(x)-f(x)}{x-0}\\ &= \lim_{x\to0}\dfrac{x|x|-0|0|}{x-0}\\ &= \lim_{x\to0}\dfrac{x|x|}{x}\\ &= \lim_{x\to0}|x| \end{align}
I know $f(x)=|x|$ is not differentiable at $0$. However, would this be? If one plugs in $0$ to $|x|$, one would get $0$, which is finite. But, if one takes the the limit from $-\infty$ and $+\infty$, the limits wouldn't exist.
Why are you mentioning $\pm\infty$? What you did is correct. Since $\lim_{x\to0}\lvert x\rvert=0$, $f'(0)=0$.