$\lim\limits_{x\to 1}\large\frac{-\sqrt{25-x^2}-(-\sqrt{24})}{x-1}$
$\Rightarrow \lim\limits_{x\to 1}\large\frac{\sqrt{24}-\sqrt{25-x^2}}{x-1}\times \frac{\sqrt{24}+\sqrt{25-x^2}}{\sqrt{24}+\sqrt{25-x^2}}$
$\Rightarrow \lim\limits_{x\to 1}\large\frac{x^2-1}{(x-1)(\sqrt{24}+\sqrt{25}-x^2)}$
$\Rightarrow \large\frac{2}{2\sqrt{24}}$
$\Rightarrow \large\frac{1}{2\sqrt 6}$
Hence (d) is the correct answer.