Let $a \in \mathbb{R}$ and prove that, using the $\epsilon-\delta$ definition of the limit, $\displaystyle \lim_{x \to a} f(x) = \infty$ iff $\displaystyle \lim_{x \to a} \frac{1}{f(x)} = 0$ and that there is $\delta > 0$ such that $f(x) > 0$ whenever $0 < |x-a| < \delta$.
For the first question, we are given that $$\forall M>0, \exists \delta \quad 0<|x-a|<\delta \implies \quad f(x) > M$$ and need to show this is equivalent to $$\forall \epsilon, \exists \delta \quad 0<|x-a|<\delta \implies \quad \left |\dfrac{1}{f(x)} \right |< \epsilon.$$ I am not to sure how to show this part. As for the second part of the question, couldn't we just have $f(x) = -5$ and that never be true?