  Let $w\in\mathbf F_k$ be a non-trivial word and denote by $w(G)\subseteq G$
the image of the associated word map $w\colon G^k\to G$. Let $G$ be one of the
finite groups ${\rm S}_n,{\rm GL}_n(q),{\rm Sp}_{2m}(q),{\rm
GO}_{2m}^\pm(q),{\rm GO}_{2m+1}(q),{\rm GU}_n(q)$ ($q$ a prime power, $n\geq
2$, $m\geq 1$), or the unitary group ${\rm U}_n$ over $\mathbb C$. Let $d_G$ be
the normalized Hamming distance resp. the normalized rank metric on $G$ when
$G$ is a symmetric group resp. one of the other classical groups and write
$n(G)$ for the permutation resp. Lie rank of $G$. For $\varepsilon>0$, we prove
that there exists an integer $N(\varepsilon,w)$ such that $w(G)$ is
$\varepsilon$-dense in $G$ with respect to the metric $d_G$ if $n(G)\geq
N(\varepsilon,w)$. This confirms metric versions of a conjectures by Shalev and
Larsen. Equivalently, we prove that any non-trivial word map is surjective on a
metric ultraproduct of groups $G$ from above such that $n(G)\to\infty$ along
the ultrafilter. As a consequence of our methods, we also obtain an alternative
proof of the result of Hui-Larsen-Shalev that $w_1({\rm SU}_n)w_2({\rm
SU}_n)={\rm SU}_n$ for non-trivial words $w_1,w_2\in\mathbf F_k$ and $n$
sufficiently large.
