I know that 0*∞ is an indeterminate form. Is this provable?
What is yourdefinition of "indefinite form"?
I suspect you mean that you can have functions f(x), g(x), f'(x), and g'(x) such that $\displaystyle \lim_{x\to a} f(x)= 0$. $\displaystyle \lim_{x\to a} g(x)= \infty$, $\displaystyle \lim_{x\to a} f'(x)= 0$, and $\displaystyle \lim_{x\to a} g'(x)= \infty$ such that $\displaystyle \lim_{x\to a} f(x)g(x)$ and $\displaystyle \lim_{x\to a} f'(x)g'= 0$ converge to different limits.
And you can prove that by showing such functions.