Answer
$\dfrac{1}{4}$
Work Step by Step
Re-arrange the equation as follows: $\lim\limits_{(x,y) \to (4,3)} \dfrac{\sqrt x-\sqrt {y+1}}{(x-y)-1}=\lim\limits_{(x,y) \to (4,3)} \dfrac{\sqrt x-\sqrt {y+1}}{(\sqrt x+\sqrt {y+1})(\sqrt x-\sqrt {y+1})}$
and $\lim\limits_{(x,y) \to (4,3)} \dfrac{\sqrt x-\sqrt {y+1}}{(\sqrt x+\sqrt {y+1})(\sqrt x-\sqrt {y+1})}=\lim\limits_{(x,y) \to (4,3)} \dfrac{1}{(\sqrt x-\sqrt {y+1})}$
Plug the limits, then we get $\dfrac{1}{(\sqrt 4+\sqrt {3+1})}=\dfrac{1}{(\sqrt 4+\sqrt {4})}=\dfrac{1}{4}$
