Asymptotic behavior of two Bernstein-type operators is studied in this paper. In the first case, the rate of convergence of a Bernstein operator for a bounded function f is studied at points x where f (x+) and f (x-) exist. In the second case, the rate of convergence of a Szasz operator for a function f whose derivative is of bounded variation is studied at points X where f (x+) and f (x-) exist. Estimates of the rate of convergence are obtained for both cases and the estimates are the best possible for continuous points.