Limits of special Weierstrass points

Let C be the union of two general connected, smooth, nonrational curves X and Y inter- secting transversally at a point P. Assume that P is a general point of X or of Y. Our main result describes all limits on C of special Weierstrass points along smooth curves degenerating to C . As an application, we recover in a unified and conceptually simpler way the computations made by Diaz and Cukierman of divisor classes of curves with special Weierstrass points in the moduli space of stable curves. In our approach there are no multiplicity issues, an usual nuisance of the method of test curves.