Let $\displaystyle f(x)=x^2\sin{1/x}$ for $\displaystyle 0<x\leq 1$ and $\displaystyle f(0)=0$ , and let $\displaystyle g(x)=x^2$ for $\displaystyle x\in [0,1]$ .

Then both f and g are differentiable on [0,1] and $\displaystyle g(x)>0$ for $\displaystyle x\neq 0$ .

Show that $\displaystyle \lim_{x \to 0} f(x) = 0 = \lim_{x \to 0} g(x)$ and that $\displaystyle \lim_{x \to 0} f(x)/g(x)$ does not exist.