# Finding the limit when one part goes to infinity and the other part goes to zero

Let's say you have a function $$f(x) = h(x)g(x)$$. You know that $h(x) \to \infty$ as $x \to \infty$, and $g(x) \to 0$ as $x \to \infty$.

How can you go about finding the limit of $f(x)$ as $x \to \infty$

-

Perhaps you should consider $\lim \dfrac{g(x)}{\frac{1}{h(x)}}$, as now the numerator and denominator both go to 0. So now, you can use L'Hopital's rule if they are differentiable.
This won't work too well if, say, $h(x)=x$ and $g(x)=e^{-x}$, but I guess we have to let Angada work out some of the answer. –  Gerry Myerson Oct 1 '11 at 1:15
Very True. ${}{}{}$ –  mixedmath Oct 1 '11 at 2:11