# Proper way to write "simultaneous approaching" the limit

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?

• The proper way is always to use words. Jan 21 at 3:53
• How about $x\rightarrow0\longleftrightarrow 0\leftarrow y$. Jan 21 at 5:05
• I agree with @Misha Lavrov. Presumably what you're writing is already heavily symbolic, and thus adding unnecessary symbolism to it overloads the reader with symbolic clutter, which tends to make your writing difficult to read. Depending on how this arises, something like "Since $x \rightarrow y$ is equivalent to $y \rightarrow x,$ it follows that $\ldots$" For example, see the 4th sentence of this answer. Jan 21 at 10:00

[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

$$y \to 0$$ as $$x \to 0$$

which is not the same as

$$y \to 0$$ if $$x \to 0$$

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.