For all positive functions $f$ and $g$ of the real variable $x$, let $\sim$ be a relation defined by
$f \sim g$ if and only if $\lim_{x \rightarrow \infty} \frac{f(x)}{g(x)} = 1$
Then if $f \sim g$, we have for example, that $f^2 \sim g^2, \sqrt{f} \sim \sqrt{g}, f+g \sim 2g$, but we do NOT have that $e^f \sim e^g$. What other operations preserve, or do not preserve the asymptotic relation $\sim$?