Let $P\subseteq \mathbb{N}$ be the set of primes, and for any integer $n>1$ let $L(n) = \max\{p \in P: p | n\}$ be the largest prime divisor of $n$. Moreover, for $n \in \mathbb{N}$ with $n>1$ we let $M(n)$ to be the median of the set $$\{L(m)/m : m\in \mathbb{N} \land 1 < m \leq n\}.$$

Does $\lim_{n\to\infty}M(n)$ exist? If yes, is its value known?

Let $P\subseteq \mathbb{N}$ be the set of primes, and for any integer $n>1$ let $L(n) = \max\{p \in P: p \mid n\}$ be the largest prime divisor of $n$. Moreover, for $n \in \mathbb{N}$ with $n>1$ we let $M(n)$ to be the median of the set $$\{L(m)/m : m\in \mathbb{N} \land 1 < m \leq n\}.$$

Does $\lim_{n\to\infty}M(n)$ exist? If yes, is its value known?