  The singular limit of the thin film Muskat problem is performed when the
density (and possibly the viscosity) of the lighter fluid vanishes and the
porous medium equation is identified as the limit problem. In particular, the
height of the denser fluid is shown to converge towards the solution to the
porous medium equation and an explicit rate for this convergence is provided in
space dimension d $\le$ 4. Moreover, the limit of the height of the lighter
fluid is determined in a certain regime and is given by the corresponding
initial condition.
