Limit of the product of these two functions

LagrangeEuler
If we have two functions ##f(x)## such that ##\lim_{x \to \infty}f(x)=0## and ##g(x)=\sin x## for which ##\lim_{x \to \infty}g(x)## does not exist. Can you send me the Theorem and book where it is clearly written that
$$\lim_{x \to \infty}f(x)g(x)=0$$
I found that only for sequences, but it should be correct for functions also.

$$\lim_{x \to \infty}f(x)g(x)=0$$