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 $\mathcal{O}(\frac{1}ε)$ comparing to the inter-transition rates among them, where $0 < ε\ll 1$. Several asymptotic results are obtained as $ε\rightarrow 0$ concerning the convergence of Kolmogorov backward equation, Poincaré constant, (modified) logarithmic Sobolev constant to their counterparts of certain reduced Markov chain. Both reversible and irreversible Markov chains are considered.