A geometric transition is a continuous path of geometries which abruptly changes type in the limit. We explore geometric transitions of the Cartan subgroup in SL(n,R). For n=3, the Cartan subgroup has precisely 5 limits, and for n=4, there are 15 limits, which give rise to generalized cusps on convex projective 3-manifolds. When n > 6, there is a continuum of non-conjugate limits of the Cartan subgroup, distinguished by projective invariants. To prove these results, we use some new techniques of working over the hyperreal numbers.