Find the hyperbolic distance between $(0; 0; 0)$ and $(0; 0; \frac12)$ in the Poincare model. Recall that the Poincare model deems $d(P_1; P_2)=\int\frac{2}{1-r^2}ds$. What about the distance between $(0; 0; 0)$ and $(0; 0; A)$ as $A$ approaches $1$?
As we integrate along a horizontal line, the path differential $ds$ consists only of a horizontal component, i.e. $dy = dz = 0$ but still stuck.