If $G(x)=-\sqrt{25-x^2}$ then $\lim\limits_{x \to 1}\large\frac{G(x)-G(1)}{x-1}$ has the value

Question

$(a)\;1/24\qquad(b)\;1/5\qquad(c)\;-\sqrt{24}\qquad(d)\;None\;of\;these$

Answer 1

$\lim\limits_{x\to 1}\large\frac{-\sqrt{25-x^2}-(-\sqrt{24})}{x-1}$

$\Rightarrow \lim\limits_{x\to 1}\large\frac{\sqrt{24}-\sqrt{25-x^2}}{x-1}\times \frac{\sqrt{24}+\sqrt{25-x^2}}{\sqrt{24}+\sqrt{25-x^2}}$

$\Rightarrow \lim\limits_{x\to 1}\large\frac{x^2-1}{(x-1)(\sqrt{24}+\sqrt{25}-x^2)}$

$\Rightarrow \large\frac{2}{2\sqrt{24}}$

$\Rightarrow \large\frac{1}{2\sqrt 6}$

Hence (d) is the correct answer.