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The purpose of this article is twofold. The first aim is to characterize an$n$-dimensional hyperbolic complex manifold $M$ exhausted by a sequence$\{\Omega_j\}$ of domains in $\mathbb C^n$ via an exhausting sequence$\{f_j\colon \Omega_j\to M\}$ such that $f_j^{-1}(a)$ converges to a boundarypoint $\xi_0 \in \partial \Omega$ for some point $a\in M$. Then, our second aimis to show that any spherically extreme boundary point must be stronglypseudoconvex.

Exhaustion of hyperbolic complex manifolds and relations to the squeezing function