  The purpose of this article is to investigate a hyperbolic complex manifold
$M$ exhausted by a pseudoconvex domain $\Omega$ in $\mathbb C^n$ via an
exhausting sequence $\{f_j\colon \Omega\to M\}$ such that $f_j^{-1}(a)$
converges to a boundary point $\xi_0 \in \partial \Omega$ for some point $a\in
M$.
