  Let C be the union of two general connected, smooth, nonrational curves X and
Y intersecting transversally at a point P. Assume that P is a general point of
X or of Y. Our main result, in a simplified way, says:
  Let Q be a point of X. Then Q is the limit of special Weierstrass points on a
family of smooth curves degenerating to C if and only if Q is not P and either
of the following conditions hold: Q is a special ramification point of the
linear system |K_X+(g_Y+1)P|, or Q is a ramification point of the linear system
|K_X+(g_Y+1+j)P| for j=-1 or j=1 and P is a Weierstrass point of Y.
  Above, g_Y stands for the genus of Y and K_X for a canonical divisor of X.
  As an application, we recover in a unified and conceptually simpler way
computations made by Diaz and Cukierman of certain divisor classes in the
moduli space of stable curves. In our method there is no need to worry about
multiplicities, an usual nuisance of the method of test curves.
