$\lim\limits_{x\to\sqrt{2}}\frac{x^2-2}{x^2-x+\sqrt{2}-2}$
$=\lim\limits_{x\to\sqrt{2}}\frac{(x-\sqrt{2})(x+\sqrt{2})}{(x^2-2)-(x-\sqrt{2})}$
$=\lim\limits_{x\to\sqrt{2}}\frac{(x-\sqrt{2})(x+\sqrt{2})}{(x-\sqrt{2})(x+\sqrt{2}-1)}$
$=\lim\limits_{x\to\sqrt{2}}\frac{x+\sqrt{2}}{x+\sqrt{2}-1}$
$=\frac{2\sqrt{2}}{2\sqrt{2}-1}=\frac{8+2\sqrt{2}}{7}$