Question

The value of $\underset{x\to a}{\lim }\sqrt{a^2-x^2}\cot \frac{\pi }{2}\sqrt{\frac{a-x}{a+x}}$ is

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

Answer 1

Answer: (C)

$\underset{x\to a}{\lim }\sqrt{a^2-x^2}\cot \frac{\pi }{2}\sqrt{\frac{a-x}{a+x}}$
$(0\times \infty$ form $)$
$=\underset{x\to a}{\lim }\frac{\sqrt{a^2-x^2}}{\tan \frac{\pi }{2}\sqrt{\frac{a-x}{a+x}}}$
$(\frac{0}{0}$ form $)$
$=\underset{x\to a}{\lim }\frac{\frac{-2x}{2\sqrt{a^2-x^2}}}{-{\sec }^2\frac{\pi }{2}\sqrt{\frac{a-x}{a+x}}\times \frac{\pi }{2}\times \frac{2a}{2(a+x)\sqrt{a^2-x^2}}}$
$=\frac{4a}{\pi }$