Define $g\colon\mathbb{R}\to\mathbb{R}$ by $g(x)=x|x|$. Prove that $g$ is differentiable at $0$.
Am I still able to use the precise definition of the limit if the function is a composition of two functions? or do I have to use the product rule?
2026-04-08 00:44:34.1775609074
Proving that $g(x)=x|x|$ is differentiable at $0$
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Yes, you are able to use the definition:$$\lim_{x\to0}\frac{g(x)-g(0)}x=\lim_{x\to0}\lvert x\rvert=0.$$