  We prove stability results for nonlinear diffusion equations of the porous
medium and fast diffusion types with respect to the nonlinearity power $m$:
solutions with fixed data converge in a suitable sense to the solution of the
limit problem with the same data as $m$ varies. Our arguments are elementary
and based on a general principle. We use neither regularity theory nor
nonlinear semigroups, and our approach applies to e.g. Dirichlet problems in
bounded domains and Cauchy problems on the whole space.
