Do MEMS accelerometers have a lower frequency limit?
There is no lower limit to the frequency response of the acceleration to output transfer function. As the other answers have said, they detect orientation, which means detecting gravity all the way down to DC.
However, be aware that all accelerometers, regardless of technology, suffer from the velocity to acceleration transfer function falling by 6dB per octave towards DC, and the position to acceleration transfer function falling by 12dB/octave, as the shaking frequency decreases. At lower frequencies, the same acceleration is going to correspond to larger amplitudes and higher velocities, so it will certainly seem like the sensitivity falls towards DC.