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