Question

Among (i) $\underset{x\to \infty }{\lim }{\sec }^{-1}\left(\frac{x}{\sin x}\right)$ and (ii) $\underset{x\to \infty }{\lim }{\sec }^{-1}\left(\frac{\sin x}{x}\right)$,

• A. (i) exists, (ii) does not exist
• B. (i) does not exist, (ii) exists
• C. Both (i) and (ii) exist
• D. Neither (i) nor (ii) exists

Answer 1

Answer: (A)

(i) $\underset{x\to \infty }{\lim }{\sec }^{-1}\left(\frac{x}{\sin x}\right)={\sec }^{-1}\left(\frac{\infty }{\sin \infty }\right)$
$={\sec }^{-1}\left(\frac{\infty }{\text{ any value between }-1\text{ to }1}\right)$
$={\sec }^{-1}(\pm \infty )=\frac{\pi }{2}$
(ii) $\underset{x\to \infty }{\lim }{\sec }^{-1}\left(\frac{\sin x}{x}\right)={\sec }^{-1}\left(\frac{\sin \infty }{\infty }\right)$
$={\sec }^{-1}\left(\frac{\text{ any value between }-1\text{ to }1}{\infty }\right)$
$={\sec }^{-1}0=$ not defined
Hence, (i) exists but (ii) does not exist.