Log of the limiting fraction of empty bins

Throw $n$ balls independently and uniformly into $n$ bins, and let $P_n$ denote the fraction of bins left empty. Find $\lim_{n\to\infty}\ln\big(\mathbb{E}[P_n]\big)$; if the limit does not exist, answer $2$.

$-1$. A given bin stays empty with probability $(1-\tfrac1n)^n$, and by linearity this is also $\mathbb{E}[P_n]$. It tends to $e^{-1}$, whose logarithm is $-1$.