  We study the set of tangent limits at a given point to a set definable in any
o-minimal structure by characterizing the set of exceptional rays in the
tangent cone to the set at that point and investigating the set of tangent
limits along these rays. Several criteria for determining exceptional rays will
be given. The main results of the paper generalize, to the o-minimal setting
and to arbitrary dimension, the main results of O'Shea--Wilson which deals with
algebraic surfaces in $\mathbb R^3$.
