The large Reynolds number asymptotic approximations of the neutral curve of Taylor–Couette flow subject to an axial uniform magnetic field are analysed. The flow has been extensively studied since the early 1990s as the magneto-rotational instability (MRI) occurring in the flow may explain the origin of the instability observed in some astrophysical objects. Elsewhere, the ideal approximation has been used to study high-speed flows, which sometimes produces paradoxical results. For example, ideal flows must be completely stabilised for a sufficiently strong applied magnetic field. On the other hand, the vanishing magnetic Prandtl number limit of the stability should be purely hydrodynamic, so instability must occur when Rayleigh’s stability condition is violated. Our first discovery is that this apparent contradiction can be resolved by showing the abrupt appearance of the hydrodynamic instability at a certain critical value of the magnetic Prandtl number. This is found using the asymptotically large Reynolds number limit but with a sufficiently long wavelength to retain some diffusive effects. Our second finding concerns the so-called Velikhov–Chandrasekhar paradox, namely the mismatch of the zero external magnetic field limit of the Velikhov–Chandrasekhar stability criterion and Rayleigh’s stability criterion. We show for fully wide-gap cases that the high Reynolds number asymptotic analysis of the MRI naturally yields the simple stability condition that describes smooth transition from Rayleigh to Velikhov–Chandrasekhar stability criteria with increasing Lundquist number.