Definition: The tangent cone to a set $S \subset \mathbb{R}^n$ at a point $x \in \mathbb{R}^n$, denoted $T_S(x)$, is the set of all vectors $w \in \mathbb{R}^n$ such that there exist $x_i \in S$, $\tau_i \in \mathbb{R}_+$ with $x_i \to x$, $\tau_i \to 0$, and $w = \lim_{i \to \infty} \frac{x_i-x}{\tau_i}$.
I have been searching the web for any fundamental properties of the tangent cone that would help with computations and failed to find anything. I would appreciate if someone could point out some literature or an answer as to how the tangent cone behaves.
For example, what can we say about the tangent cone of a cartesian product of two subsets of euclidean spaces $T_{S_1 \times S_2}(x_1,x_2)$ with $S_1 \subset \mathbb{R}^{n_1}, S_2 \subset \mathbb{R}^{n_2}$. What about disjoint unions of sets (i.e., $T_{S_1 \cup S_2}(x)$ where $S_1 \cap S_2 = \emptyset$).