Undergrad 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.

 

[PeroK]

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.

If $g$ is any bounded function then there is a straightforward epsilon-delta proof.

 

[mathman]

|f(x)g(x)| is bounded by |f(x)| goes to 0

 

[Twisted_Alexander]

I don't know of a specific place that states this exact problem, but this is an application of the Squeeze theorem which can be found in any calculus textbook.