If we know that $\lim_{x\to a}f(x)=\infty$, and $\lim_{x\to a}g(x)=0$, then what will be $\lim_{x\to a}(f(x)g(x))$? How do you prove the result?
By using examples I can see that the limit can be $\infty$, $-\infty$, $0$ and any other real number depending what your $f(x)$ and $g(x)$ are but I am not sure how to construct a proof-like argument here.
If you want an example where $f(x)\to\infty$, $g(x)\to0$ and $f(x)g(x)$ tends to a given real number $c$ as $x\to a$, just take $$f(x)=\frac{1}{(x-a)^2}\quad\hbox{and}\quad g(x)=c(x-a)^2\ .$$