The dependence of the measured damping rate upon the Neutral Beam Heating power (PNBI) and plasma β has been experimentally investigated for n = 1 Toroidal Alfvén Eigenmodes (TAEs) in JET limiter plasmas with a monotonic safety factor profile and low edge magnetic shear. The mode frequency is observed to decrease with increasing Beta, as predicted by various theoretical models. For low PNBI<3MW, a single n = 1 TAE splits into two modes closely spaced around the expected TAE frequency, and the mode damping rate decreases with increasing β. On the other hand, at higher PNBI>6MW, no frequency splitting is observed and there are indications that the mode damping rate increases with increasing β.