Let $\displaystyle f(x) = x^2\sin{1/x}$ for $\displaystyle x\neq 0$ , let $\displaystyle g(x) = \sin{x}$ for $\displaystyle x\in\mathbb{R}$ . Show that $\displaystyle \lim_{x\to 0 }f(x)/g(x) = 0$ but that $\displaystyle \lim_{x\to 0}f'(x)/g'(x)$ does not exist.