  Perturbation theory is an indispensable tool for studying the cosmic
large-scale structure, and establishing its limits is therefore of utmost
importance. One crucial limitation of perturbation theory is shell-crossing,
which is the instance when cold-dark-matter trajectories intersect for the
first time. We investigate Lagrangian perturbation theory (LPT) at very high
orders in the vicinity of the first shell-crossing for random initial data in a
realistic three-dimensional Universe. For this we have numerically implemented
the all-order recursion relations for the matter trajectories, from which the
convergence of the LPT series at shell-crossing is established. Convergence
studies performed up to the 40th order reveal the nature of the
convergence-limiting singularities. These singularities are not the well-known
density singularities at shell-crossing but occur at later times when LPT
already ceased to provide physically meaningful results.
