  The abelianization is a functor from groups to abelian groups, which is left
adjoint to the inclusion functor. Being a left adjoint, the abelianization
functor commutes with all small colimits. In this paper we investigate the
relation between the abelianization of a limit of groups and the limit of their
abelianizations. We show that if $\mathcal{T}$ is a countable directed poset
and $G:\mathcal{T}\longrightarrow\mathcal{G} rp$ is a diagram of groups that
satisfies the Mittag-Leffler condition, then the natural map
$$\mathrm{Ab}(\lim_{t\in\mathcal{T}}G_t)\longrightarrow\lim_{t\in\mathcal{T}}\mathrm{Ab}(G_t)$$
is surjective, and its kernel is a cotorsion group. In the special case of a
countable product of groups, we show that the Ulm length of the kernel does not
exceed $\aleph_1$.
