  A surface in the Lorentz-Minkowski $3$-space is generally a mixed type
surface, namely, it has the lightlike locus. We study local differential
geometric properties of such a locus on a mixed type surface. We define a frame
field along a lightlike locus, and using it, we define two lightlike ruled
surfaces along a lightlike locus which can be regarded as lightlike
approximations of the surface along the lightlike locus. We study a
relationship of singularities of these lightlike surfaces and differential
geometric properties of the lightlike locus. We also consider the intersection
curve of two lightlike approximations, which gives a model curve of the
lightlike locus.
