We consider the fast diffusion equation in a bounded smooth domain with homogeneous Dirichlet conditions. It is known that bounded positive solutions of such problem extinguish in a finite time T, and also that such solutions approach a separate variable solution provided the parameter m appearing in the equation is in a suitable range. Here we are interested in describing the behaviour of the solutions near the extinction time. We first show that, for a certain range of the parameter m appearing in the equation, the convergence takes place uniformly in the relative error norm. Then, we study the question of rates of convergence. For m close to 1 we get such rates by means of entropy methods and weighted Poincarè inequalities. The analysis of the latter point makes an essential use of fine properties of an associated stationary elliptic problem when m tends to 1, and such a study has an independent interest. This is a joint work with Matteo Bonforte and Juan Luis Vazquez (UAM, Madrid).