This note is first concerned with the approximation of the solution operators for initial-value problems, and that in case the solution operators have a certain smoothing property similar to that of holomorphic semigroups. A convergence theorem with rates given by Peetre-Thomée (and later by Löfström) in the setting of concrete (Besov) spaces is reproduced in the frame of arbitrary Banach spaces. Secondly it is shown that the stable limit laws of probability theory may be seen as an application of this theorem of numerical analysis. Moreover, one obtains sharper rates of convergence for these limit laws.