  We propose a systematic and topological study of limits $\lim_{\nu\to
0^+}G_\mathbb{R}\cdot(\nu x)$ of continuous families of adjoint orbits for
non-compact simple Lie groups. This limit is always a finite union of nilpotent
orbits. We describe explicitly these nilpotent orbits in terms of Richardson
orbits in the case of hyperbolic semisimple elements. We also show that one can
approximate minimal nilpotent orbits or even nilpotent orbits by elliptic
semisimple orbits. The special cases of $\mathrm{SL}_n(\mathbb{R})$ and
$\mathrm{SU}(p,q)$ are computed in detail.
