I have two functions, $f(x)$ and $g(x)$ that are continuous on the real number line. I'm taking the limit of their product:
$\lim_{x\rightarrow\infty}f(x)g(x)$
Suppose that the limit $\lim_{x\rightarrow\infty}g(x)=c$.
Is there a continuous function $f(x)$ such that:
$\lim_{x\rightarrow\infty}f(x)g(x)\neq\lim_{x\rightarrow\infty}cf(x)$