  The large Reynolds number asymptotic approximation of the neutral curve of
Taylor-Couette flow subject to axial uniform magnetic field is analysed. The
flow has been extensively studied since early 90's as the magneto-rotational
instability (MRI) occurring in the flow could possibly explain the origin of
the instability observed in certain astrophysical objects. Elsewhere the ideal
approximation has been used to study high-speed flows, whilst it sometimes
produces paradoxical results. For example, ideal flows must be completely
stabilised for strong enough applied magnetic field, but 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. The first our discovery is that this apparent contradiction can be
resolved by showing the abrupt appearance of the hydrodynamic instability at
certain critical value of magnetic Prandtl number, which can be found by
asymptotically large Reynolds number limit but with long enough wavelength to
retain some diffusive effects. The second our finding concerns 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.
