  The asymptotic behaviour of the components of the Weyl tensor and of the
energy-momentum tensor in the Penrose limit is determined. In both cases a
peeling-off property is found. Examples of different types of matter are
provided. The expansion and shear of the congruence of null geodesics along
which the Penrose limit is taken are determined. Finally, the approach to the
singularity in the Penrose limit of cosmological space-times is discussed.
