  We consider continuous-time Markov chain on a finite state space X. We assume
X can be clustered into several subsets such that the intra-transition rates
within these subsets are of order $\mathcal{O}(\frac{1}{\epsilon})$ comparing
to the inter-transition rates among them, where $0 < \epsilon \ll 1$. Several
asymptotic results are obtained as $\epsilon \rightarrow 0$ concerning the
convergence of Kolmogorov backward equation, Poincar\'e constant, (modified)
logarithmic Sobolev constant to their counterparts of certain reduced Markov
chain. Both reversible and irreversible Markov chains are considered.
