Kalman's formal limiting procedure is applied to some recent results in sequential discrete-time non-linear filtering and smoothing to obtain the corresponding estimation algorithms for continuous-time non-linear dynamic systems. The resulting filtering algorithm is found to agree with the well-known Detchmendy-Sridhar filter which was obtained via another method. The present smoothing algorithm is a new result. It is argued that the combined filter-smoothing results here lead to an estimation algorithm which is second order in both system dynamics and measurement function non-linearity.