Given two functions that do not converge as $x\to\infty$. Can their product converge?

Question

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?

Answer 1

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 $x \to \infty$.