The constraint of incompressibility is often used to simplify the magnetohydrodynamic (MHD) description of linearized plasma dynamics because it does not affect the ideal MHD marginal stability point. In this paper two methods for introducing incompressibility are compared in a cylindrical plasma model: in the first method, the limit γ → ∞ is taken, where γ is the ratio of specific heats; in the second, an anisotropic mass tensor ρ is used, with the component parallel to the magnetic field taken to vanish, ρ → 0. Use of resistive MHD reveals the nature of these two limits because the Alfvén and slow magnetosonic continua of ideal MHD are converted to point spectra and moved into the complex plane. Both limits profoundly change the slow magnetosonic spectrum, but only the second limit faithfully reproduces the resistive Alfvén spectrum and its wavemodes. In ideal MHD, the slow magnetosonic continuum degenerates to the Alfvén continuum in the first method, while it is moved to infinity by the second. The degeneracy in the first is broken by finite resistivity. For numerical and semi-analytical study of these models, we choose plasma equilibria which cast light on puzzling aspects of results found in earlier literature.