This is the entire proof for the 0/0 case and I am very lost. I understand choosing $r>A$, but I don't understand why we're choosing $q$. I can see why 18 is true, but I do not understand how the strict inequality becomes a weak inequality going from 18 to 19. Finally, and perhaps most importantly, I don't understand how 19 shows that $f(x)/g(x)\rightarrow A$ as $x\rightarrow a$.
Can someone help me follow this through?