Why map $x\to \frac{x}{|x|^2}$ map disc to half space?

Question

I was reading Gilberg And Trudinger. In that one of the proof Author mentioned that

map $x\to \frac{x}{|x|^2}$ map disc to half space.

I tried with $(\pm 1/n,0)$ and $(0,\pm 1/n)$ It gives points form whole $Z\times Z$ which contradicts with Auther claim. Where I am missing? Please Help me

Any Help will be appreciated