STEPS: Proving $\displaystyle \lim_{x\to a} f(x)g(x) = -\infty$ Given $\displaystyle \lim_{x \to a} f(x) = \infty$ and $\displaystyle\lim_{x \to a} g(x) = c$ where $c<0$.
Prove that $\displaystyle \lim_{x \to a} f(x)g(x) = -\infty$ only using the precise definitions of limit and infinite limit.
I get the intuitive idea. But... how do we actually write it out?

Let $A>0$ and $\epsilon=-\frac c2>0$ then there's $\delta_1,\delta_2>0$ such that 
$$|x-a|<\delta_1\implies f(x)>A$$
and
$$|x-a|<\delta_2\implies \frac{3c}2<g(x)<\frac c2$$
Now we have
$$|x-a|<\delta=\min(\delta_1,\delta_2)\implies f(x)g(x)<\frac{Ac}2$$
and this for arbitrary $A$. We conclude.
