Say there is some correspondence between $x$ and $y$. If we know that $\lim\limits_{x\rightarrow0}y=0$, we can safely say that "$y$ approaches zero if $x$ approaches zero" without causing confusion. It also seems that the notation $$y\rightarrow0\quad(x\rightarrow0)$$ is accepted as a convention in formal writings, which indeed is not ambiguous at all.
However, if we both have $$y\rightarrow0\quad(x\rightarrow0)$$ and $$x\rightarrow0\quad(y\rightarrow0)$$ is it acceptable to write $$y\rightarrow0\Leftrightarrow x\rightarrow0$$ ?
Usually $P\Leftrightarrow Q$ is understood as a logical assertation involving the truth of propositions $P$ and $Q$. However, here $y\rightarrow0$ and $x\rightarrow0$, when isolated, are not propositions. Hence, I'm concerned whether "$\Leftrightarrow$" is by convention acceptable in a formal context or being abused. If it is not acceptable, is there a concise but proper way to make the assertion?
[This should probably be a comment, but it's a little long and needs formatting, so here it is as an answer.]
If you write $$ y \to 0 \quad ( x \to 0 ) $$ out in words (well, a little bit in words), then it becomes
which is not the same as
So I don't think that you should use a symbol that means ‘if and only if’. You really want ‘as and only as’.
If you can find a symbol that would go well between $ y \to 0 $ and $ x \to 0 $ (rather than using a space and parentheses), then maybe it can be adapted here. Otherwise, I agree with Misha's comment: use words.