Let us consider the set of differentiable real functions $\mathcal{F}$. On it we can consider the equivalence relation $$f\sim g \Leftrightarrow \lim_{x\to\infty}\frac{f(x)}{g(x)}=1.$$ The equivalence classes of the respective quotient are denoted as $o(f)$, where $f$ is a representative.
This notation is often used, by I have never seen a systematic treatment of the set by itself even though it seems interesting. For example it is a group under multiplication, $o(1)$ being the identity, and I guess it could be turned into a ring if we consider also composition. Also it can inherit the quotient topology from $\mathcal{F}$, it is a real vector space, and it has a total order given by $o(f)\geq o(g) \iff \lim f/g\geq 1$.
Has this set been studied somewhere under any name? Does it have notable properties?