Limits of tangent spaces to definable sets

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 $\mathbb R^3$.