Choosing Strips

Two players $A$ and $B$ play the following game:

$A$ divides an $n \times n$ square into strips of unit width (and various integer lengths). After that, player $B$ picks an integer $k\ (1 \leq k \leq n)$ and removes all strips of length $k.$ Let $l(n)$ be the largest area that $B$ can remove regardless of how $A$ divides the square into strips.

Evaluate $\displaystyle \lim_{n \to \infty} \frac{l(n)}{n}.$

Proposed by Iurie Boreico