On notation for function mapping

Question

How can I write the following multiple mappings in a more compact and elegant way?

$$ \begin{align} \bar \mu \leq 0&: \lim \limits_{\rho \to \infty } \mu \to 0 \\ \bar \mu \geq 1&: \lim \limits_{\rho \to \infty } \mu \to 1 \\ \bar \mu \in (0,1)&: \lim\limits_{\rho \to \infty } \mu \to \bar \mu \\ \end{align} $$

Answer 1

Your notation makes little sense to me, but I think what you are trying to say can be written as follows: $$ \lim_{\rho\to\infty}\mu=\begin{cases}0&\bar\mu\le0\\1&\bar\mu\ge1\\\bar\mu&\bar\mu\in(0,1)\end{cases}$$