  The excitation conditions of the magnetorotational instability are studied
for axially unbounded Taylor-Couette flows of various gap widths between the
cylinders. The cylinders are considered as made from both perfect-conducting or
insulating material and the conducting fluid with a finite but small magnetic
Prandtl number rotates with a quasi-Keplerian velocity profile. The solutions
are optimized with respect to the wave number and the Reynolds number of the
rotation of the inner cylinder. For the axisymmetric modes we find the critical
Lundquist number of the applied axial magnetic field the lower the wider the
gap between the cylinders. A similar result is obtained for the induced cell
structure: the wider the gap the more spherical the cells are. The marginal
rotation rate of the inner cylinder -- for fixed size of the outer cylinder --
always possesses a minimum for not too wide and not too narrow gap widths. For
perfect-conducting walls the minimum lies at $r_{\rm in}\simeq 0.4$ while it is
at $r_{\rm in}\simeq 0.5$ for insulating walls where $r_{\rm in}$ is the
normalized radius of the inner cylinder. The lowest magnetic field amplitudes
to excite the instability are required for Taylor-Couette flows between
perfect-conducting cylinders with gaps corresponding to $r_{\rm in}\simeq 0.2$.
For even wider and also for very thin gaps the needed magnetic fields and
rotation frequencies are shown to become rather huge.
  Also the nonaxisymmetric modes with $|m|=1$ have been considered. Their
excitation generally requires stronger magnetic fields and higher magnetic
Reynolds numbers in comparison to those for the axisymmetric modes which is
true for wide-gap containers with $r_{\rm in} \lesssim 0.3$.
