quantum field theory - Conformal compatification of Minkowski and AdS

How do I show that the compactification of Minkowski is given by the quadric $$uv-\eta_{ij}x^{i}x^{j}=0$$ with an overall scale equivalence in the coordinates.I get that for $v \neq 0$, the surface can be parametrized with the Minkowski coordinates. Now for $v=0$, I can have arbitrary values of $u$, which means basically two values, $u=0$ and $u\neq 0$. So are the infinities mapped to these points ? After that is it obvious that the conformal group acts on the space time defined by the quadric ?

From, $$uv-\eta_{ij}x^{i}x^{j}=1$$ if I have to show that the boundary of $AdS_{d+1}$ is Minkowski in $d$ dimensions, how do I take the limit ?