The purpose of this article is to investigate a hyperbolic complex manifold
M
M
exhausted by a pseudoconvex domain
Ω
\Omega
in
C
n
\mathbb {C}^n
via an exhausting sequence
{
f
j
:
Ω
→
M
}
\{f_j\colon \Omega \to M\}
such that
f
j
−
1
(
a
)
f_j^{-1}(a)
converges to a boundary point
ξ
0
∈
∂
Ω
\xi _0 \in \partial \Omega
for some point
a
∈
M
a\in M
.