Let (fn) be a sequence of real-valued functions converging pointwise to a function f on a closed and bounded interval (a,b) and suppose that each of the functions fn is Riemann integrable on (a,b). It doesn't always follow that the limit functions f is Riemann integrable on (a,b). Even if f is Riemann integrable on (a,b), the equation f=lim fn may not hold. The aim of this paper is to find futher conditions on the sequence (fn) such that the limit function f is Riemann integrable and the equation (f=lim(fn holds.