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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 $x \to \infty$

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up vote 1 down vote accepted

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.

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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

Are you familiar with l'hopital's rule? If so, do you see how to rewrite your function so that it is in a form which meets the l'hopital hypotheses?

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