  For a given self-adjoint first order elliptic differential operator on a
closed smooth manifold, we prove a list of results on when the delocalized eta
invariant associated to a regular covering space can be approximated by the
delocalized eta invariants associated to finite-sheeted covering spaces. One of
our main results is the following. Suppose $M$ is a closed smooth spin manifold
and $\widetilde M$ is a $\Gamma$-regular covering space of $M$. Let $\langle
\alpha \rangle$ be the conjugacy class of a non-identity element $\alpha\in
\Gamma$. Suppose $\{\Gamma_i\}$ is a sequence of finite-index normal subgroups
of $\Gamma$ that distinguishes $\langle \alpha \rangle$. Let $\pi_{\Gamma_i}$
be the quotient map from $\Gamma$ to $\Gamma/\Gamma_i$ and $\langle
\pi_{\Gamma_i}(\alpha) \rangle$ the conjugacy class of $\pi_{\Gamma_i}(\alpha)$
in $\Gamma/\Gamma_i$. If the scalar curvature on $M$ is everywhere bounded
below by a sufficiently large positive number, then the delocalized eta
invariant for the Dirac operator of $\widetilde M$ at the conjugacy class
$\langle \alpha \rangle$ is equal to the limit of the delocalized eta
invariants for the Dirac operators of $M_{\Gamma_i}$ at the conjugacy class
$\langle \pi_{\Gamma_i}(\alpha) \rangle$, where $M_{\Gamma_i}= \widetilde
M/\Gamma_i$ is the finite-sheeted covering space of $M$ determined by
$\Gamma_i$. In another main result of the paper, we prove that the limit of the
delocalized eta invariants for the Dirac operators of $M_{\Gamma_i}$ at the
conjugacy class $\langle \pi_{\Gamma_i}(\alpha) \rangle$ converges, under the
assumption that the rational maximal Baum-Connes conjecture holds for $\Gamma$.
