trix Mn = An Cn Dn Bn , where An,Bn are two sequences of closed linear operators and Cn,Dn are assumed to be sequences of bounded linear operators acting in X. In the present paper, we determine the conditions under which we have Mn converges in the generalized sense. The primary question which has attractedourattentioniswhetherornotthereexistsarelationshipbetween the Weyl essential pseudospectrum of Mn converging in the generalized sense and that of its limit.