The radiation pressure of light can act to damp and cool the vibrational motion of a mechanical resonator, but even if the light field has no thermal component, shot noise still sets a limit on the minimum phonon occupation. In optomechanical sideband cooling in a cavity, the finite off-resonant Stokes scattering defined by the cavity linewidth combined with shot noise fluctuations dictates a quantum backaction limit, analogous to the Doppler limit of atomic laser cooling. In our work, we sideband cool a micromechanical membrane resonator to the quantum backaction limit. Monitoring the optical sidebands allows us to directly observe the mechanical object come to thermal equilibrium with the optical bath. This level of optomechanical coupling that overwhelms the intrinsic thermal decoherence was not reached in previous ground-state cooling demonstrations.