Question

The value of $\displaystyle\lim _{x \rightarrow a} \sqrt{a^{2}-x^{2}} \cot \frac{\pi}{2} \sqrt{\frac{a-x}{a+x}}$ is

• A. $\frac{2 a}{\pi}$
• B. $-\frac{2 a}{\pi}$
• C. $\frac{4 a}{\pi}$
• D. $-\frac{4 a}{\pi}$

Answer 1

Answer: (C)

$\displaystyle\lim _{x \rightarrow a} \sqrt{a^{2}-x^{2}} \cot \frac{\pi}{2} \sqrt{\frac{a-x}{a+x}}$ $(0 \times \infty$ form $)$
$=\displaystyle\lim _{x \rightarrow a} \frac{\sqrt{a^{2}-x^{2}}}{\tan \frac{\pi}{2} \sqrt{\frac{a-x}{a+x}}}$ $\left(\frac{0}{0}\right.$ form $)$
$=\displaystyle\lim _{x \rightarrow a} \frac{\frac{-2 x}{2 \sqrt{a^{2}-x^{2}}}}{-\sec ^{2} \frac{\pi}{2} \sqrt{\frac{a-x}{a+x}} \times \frac{\pi}{2} \times \frac{2 a}{2(a+x) \sqrt{a^{2}-x^{2}}}}$
$=\frac{4 a}{\pi}$