Q229 / 270 · NDA 2017 (Phase 1) · Mathematics

$\operatorname{If} F(x)=\sqrt{9-x^2}$, then what is $\lim _{x \rightarrow 1} \frac{F(x)-F(1)}{x-1}$ equal to?

  • A
    $-\frac{1}{4 \sqrt{2}}$
  • B
    $\frac{1}{8}$
  • Correct answer
    C
    $-\frac{1}{2 \sqrt{2}}$
  • D
    $\frac{1}{2 \sqrt{2}}$

Solution

$\lim _{x \rightarrow 1} \frac{F(x)-F(1)}{x-1}$ We know, $\lim _{x \rightarrow a} \frac{F(n)-F(a)}{x-a}=F^{\prime}(a)$ $\begin{aligned} & \therefore F^{\prime}(x)=\frac{d}{d x}\left(\sqrt{9-x^2}\right) \\ & =\frac{1(0-2 x)}{2 \sqrt{9-x^2}}=\frac{-x}{\sqrt{9-x^2}} \\ & F^{\prime}(1)=\frac{-1}{\sqrt{9-1}}=\frac{-1}{\sqrt{8}}=\frac{-1}{2 \sqrt{2}} \end{aligned}$

Hence, the correct answer is option: 'C'.

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