Here are two asymptotic symbols: $\asymp$ and $\sim$. I know what their mathematical definitions are, namely $f(x)\asymp g(x)$ if there exist constants $A,B$ such that $Af(x)\le g(x)\le Bf(x)$, and $f(x)\sim g(x)$ if $\lim_{x\to\infty}f(x)/g(x)=1$.
What I am not sure about is what these symbols are called. They are both asymptotic by definition but I have no idea what their nomenclature is.
The relation $f \sim g$ is usually called asymptotic equivalence; see e.g. here