I am struggling to understand how to use the definition: The vector d is said to be a tangent (or tangent vector) to 6 at a point x if there are a feasible sequence ${z_k}$ approaching x and a sequence of positive scalars $t_k$ with $t_k\rightarrow 0$ such that $\frac{z_k-x}{t_k}=d$.
I am trying to use this one the set given by the constraints $c_1 = y-x$ and $c_2 = x^3-y^4$. Is the point to just kinda see which sequence will reasult in a point and not diverge? Like in this case for the first constraint the sequence $z_k=(1/k,1/k), t_k=1/k$ works and gives (0,0), while for the second constraint the solution provides $z_k = (k^{-1/3}, k^{-1/4}), t_k =k^{-1/4} $ which gives (0,1) but then just concludes with that the tangent cone is characterized with $d_2\geq d_1 \geq 0$?
So basically, is the point of the definition to find SOME sequence that converges to some direction along each of constraints, like anything that gives some d? But then, how does one actually use those tangent vectors to find the whole cone?