We use the short-wavelength analysis to study the azimuthal magnetorotational instability (AMRI), the instability of rotating flows of an electrically conducting fluid, subject to the azimuthal magnetic field, to three-dimensional disturbances. The WKB method is used to the extended Hain-Lust equation, whereby we can retain all the terms, being otherwise liable to be missed for non-axisymmetric disturbances. For the inductionless limit, i.e. the electric resistivity is assumed much larger than viscosity, we find a new wave which requires Rossby number beyond the Liu’s limit and is unstable for the Keplerian rotation subject to current-free magnetic field. This instability needs a constraint that qr < 3~(1/2) where q is the radial wavenumber and r is the local radius. When qr >3~(1/2), we compare some results with the global analysis results.
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