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Suppose the functions are named $f(x)$ and $g(x)$ and neither converges as $x\to\infty$. Is it possible, that the product $f(x)g(x)$ converges?

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up vote 4 down vote accepted

Yes. Let for example $f,g \colon \mathbb R \to \mathbb R$ defined by $f(x) = 2 + \sin x$ and $g(x) = \frac 1{2 + \sin x}$. Then $f(x)g(x) \to 1$ for $x \to \infty$.

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Great! Thx :) I just couldn't find a proper example by myself. –  iolo Nov 9 '12 at 15:26
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