  We study the limiting behaviour of Darboux and Calapso transforms of
polarized curves in the conformal n-dimensional sphere, when the polarization
has a pole of first or second order at some point. We prove that for a pole of
first order, as the singularity is approached all Darboux transforms converge
to the original curve and all Calapso transforms converge. For a pole of second
order, a generic Darboux transform converges to the original curve while a
Calapso transform has a limit point or a limit circle, depending on the value
of the transformation parameter. In particular, our results apply to Darboux
and Calapso transforms of isothermic surfaces when a singular umbilic with
index 1/2 or 1 is approached along a curvature line.
