We show how a singular perturbation technique based on the introduction of properly scaled variables enables us to derive the asymptotic properties of coupled Langevin equations in the limit of weak noise. This technique can be applied when the macroscopic steady state is asymptotically or marginally stable. In the close vicinity of a cusp bifurcation point, a simple prescription for the adiabatic elimination of the fast variable is established. The critical variable exhibits amplified non-Gaussian fluctuations on a slow time scale. The properties of the fast variable depend on the nonlinearity of the system under consideration. Because of its coupling to the critical variable, it may exhibit amplified fluctuations of non-Gaussian nature.