  The long time behavior of an absorbed Markov process is well described by the
limiting distribution of the process conditioned to not be killed when it is
observed. Our aim is to give an approximation's method of this limit, when the
process is a 1-dimensional It\^o diffusion whose drift is allowed to explode at
the boundary. In a first step, we show how to restrict the study to the case of
a diffusion with values in a bounded interval and whose drift is bounded. In a
second step, we show an approximation method of the limiting conditional
distribution of such diffusions, based on a Fleming-Viot type interacting
particle system. We end the paper with two numerical applications : to the
logistic Feller diffusion and to the Wright-Fisher diffusion with values in
$]0,1[$ conditioned to be killed at 0.
