Question

The value of $\displaystyle\lim _{x \rightarrow \infty} arc \cos \left(\frac{arc\sin \left(\frac{\pi}{x}\right)+arc\tan \left(\frac{x}{\pi}\right)}{x \sin \frac{\pi}{x}}\right)$ is equal to

• A. $\frac{\pi }{3}$
• B. $\frac{\pi }{6}$
• C. $\frac{\pi }{4}$
• D. $\frac{\pi }{2}$

Answer 1

Answer: (A)

$cos^{- 1}\left(\frac{\left(sin\right)^{- 1} \frac{\pi }{x} + \left(tan\right)^{- 1} \frac{x}{\pi }}{x \left(\frac{sin \frac{\pi }{x}}{\left(\frac{\pi }{x}\right)}\right) \cdot \frac{\pi }{x}}\right)$
$=cos^{- 1}\left(\frac{0 + \frac{\pi }{2}}{1 \cdot \pi }\right)=cos^{- 1}\left(\frac{1}{2}\right)=\frac{\pi }{3}$ .