Sorry, as this is a rather trivial question that I am misunderstanding, but I do not understand how the distinguished point is defined. We define it as a homomorphism from some semigroup $S_{\sigma}$ to either 1 or 0 depending on whether the point is contained in $S_{\sigma} \cap \sigma^{\perp}$.
Now, consider the fan $\Sigma$ with $P^{2} \simeq X_{\Sigma}$ with the usual fan as depicted here: https://rigtriv.files.wordpress.com/2008/10/p2.jpg
The limit point is supposed to coincide with the distinguished point (cf Cox, Little, Schenck, Prop 3.2.2). So for example, the limit point for the first quadrant is (1,0,0). If we take some point contained in the first quadrant, say (1,1,1), this point should be taken to 0 by the distinguished point since it is clearly not contained in $\sigma^{\perp}$. How does (1,0,0) take this point to 0 then? Similarly, I'm having trouble seeing this with the limit points for the other 6 cones.
Thanks