I am trying to solve a set of problems, this one is causing my some troubles. For the first one I tried to use the $\epsilon-\delta$ definition but I couldn't solve it, I would appreciate some hints for it.
Let $f:(0,\infty) \to \mathbb{R}$ be a differentiable function such that f' is continuous and $f(x) > 0$ for all $x \in (0,\infty)$. Prove or give a counter example for each of the following statements:
if $\displaystyle\lim_{x\to 0^{+}}f(x)=0$ then \displaystyle\lim_{x\to 0^{+}}f'(x) exists.
if $\displaystyle\lim_{x\to\infty}f(x)=0$ then \displaystyle\lim_{x\to\infty}f'(x) = 0