Notation for an asymptotic binary relation

Question

Is there any standard notational convention for an asymptotic binary relation $R$ between two functions $f(x)$ and $g(x)$ where $f(x)=_{R}g(x)$ if and only if $\lim_{x\rightarrow \infty}\left|f(x)-g(x)\right|=0$? I have to use such a relation often, and the ordinary asymptotic equivalence relation $f(x)\sim g(x)$ is not sufficient for my proof.

Answer 1

The expression

$$ f(x) = g(x) + o(1) $$

is equivalent to

$$ \lim |f(x) - g(x)| = 0, $$

and of course the limit needs to be supplied in both cases (e.g. $x \to 0$, or $x \to \infty$, etc.).

Answer 2

If you can't find a standard notation, and I can't think of one right now, you could define a relation and call it $f(x)\sim_0 g(x) $, or something like it, to be what you want. Note the "$_0$".