I've come across two definitions of Bouligand Tangent cones.
Let $X$ be a Banach space and $S \subseteq X$ be a closed set and $x \in S$ be a point.
In F.H. Clarke's "Nonsmooth Analysis and Control Theory", Bouligand tangent cone is defined to be \begin{align*} T_S^1(x) = \left\{ \lim_{j \to \infty} \frac{x_j - x}{t_j} \colon \{x_j\} \subseteq S, \; x_j \to x, \; t_j \downarrow 0 \right\}. \end{align*} In Ozman Guler's "Foundations of Optimization" and Johannes Jahn's "Introduction to the Theory of Nonlinear Optimization", the definition is \begin{align*} T_S^2(x) = \left\{ \lim_{j \to \infty} \alpha_j (x_j - x) \colon \{x_j\} \subseteq S, \; x_j \to x, \; \{\alpha_j\} \subset \mathbb{R}_{+} \right\}. \end{align*} Clearly $T_S^1(x) \subseteq T_S^2(x)$.
My Questions are:
(1) It seems to me the reverse should not hold but I could not come up an example and which definition is more widely used?
(2) In Ozman Guler's 'Foundations of Optimization', in one proof, he freely uses the fact: if $0 \in S$ and $d \in T_S(0)$, then $ td + o(t) \in S$. I would agree this if he is using the first definition $T_S^1(0)$, that is, if we let $d_j = x_j/t_j$, then $t_j d_j \in S$ and $t_j \downarrow 0$ so that $td + o(t) \in S$. Am I missing something here? Thanks.
Both definitions are equivalent. First, note that both sets contain $0$ since it's possible that $x_j=x$. So it suffices to consider nonzero elements of these sets. Since $x_j\to x$, the only way that the limit $$\lim_{j \to \infty} \alpha_j (x_j - x)$$ can be nonzero is if $\alpha_j\to +\infty$ (recall $\alpha_j$ is assumed positive). Choosing an increasing subsequence of $\alpha_j$ (which doesn't change the limit) and replacing $\alpha_j$ by $1/t_j$, we get the first definition.
In other words, the requirement that $\alpha_j>0$ in the second definition could be replaced with $\alpha_j\to +\infty$.
This should answer your question (2) as well.