Certain analytic techniques have proved effective in the study of averages for random discrete structures and algorithms. Typically one considers a generating function for the sequence of averages. One then sets up a functional equation (usually arising from a recurrence) and solves it asymptotically by a tool-kit that involves a variety of integral transforms. The behavior of the generating function near its dominant singularities captures the asymptotic nature of the averages. (Typically, non-dominant singularities contribute periodic oscillations.) To address distributions instead of only average-case analysis, these have been extended to bivariate generating functions. One still considers dominant singularities, which are now functions of a second variable. The analysis is most informative in the neighborhood of certain values for this second variable. The analysis therefore is viewed as a perturbation. We illustrate this route by a problem that arises in sorting algorithms. To find the limit distribution of a sum of dependent random variables, the analysis involves perturbation of Rice's integration method which will be explained in a tutorial introduction.