It has been conjectured for many years that plane Couette flow is stable to infinitesimal disturbances although this has never been proved. In this paper we use a, combination of asymptotic analysis and numerical computation to examine the associated Orr-Sommerfeld differential problem in a systematic manner. We obtain new evidence that the conjecture is, in all probability, correct. In particular we find that, at a fixed large value of the Reynolds number R, as in an experiment, if a disturbance of wavenumber α has a damping rate - αci then – ci has a minimum value of order R−½ when α is of order R½. We believe that this result may be an essential prerequisite to an understanding of the stability of plane Couette flow to finite-amplitude disturbances.