This paper is concerned with approximations to nonlinear filtering problems that are of interest over a very long time interval. Since the optimal filter can rarely be constructed, one needs to compute with numerically feasible approximations. The signal model can be a jump-diffusion or just a process that is approximated by a jump-diffusion. The observation noise can be either white or of wide bandwidth. The observations can be taken in either discrete or continuous time. The cost of interest is the pathwise error per unit time over a long time interval. It is shown under quite reasonable conditions on the approximating filter and the signal and noise processes that (as time, bandwidth, process, and filter approximation, etc.) go to their limit in any way at all, the limit of the pathwise average costs per unit time is just what one would get if the approximating processes were replaced by their ideal values and the optimal filter were used. Analogous results are obtained (with appropriate scaling) if the observations are taken in discrete time, and the sampling interval also goes to zero. For these cases, the approximating filter is a numerical approximation to the optimal filter for the presumed limit (signal, observation noise) problem.