Question

$\underset{x\to \frac{\pi }{2}}{lim}\frac{\sin x}{{\cos }^{-1}\left[\frac{1}{4}(3\sin x-\sin 3x)\right]}$ where [] denotes greatest integer function is

• A. $\frac{2}{\pi }$
• B. 1
• C. $\frac{4}{\pi }$
• D. does not exist

Answer 1

Answer: (A)

$\frac{2}{\pi }$

$L=\underset{x\to \frac{\pi }{2}}{lim}\frac{\sin x}{{\cos }^{-1}\left[\frac{1}{4}(3\sin x-\sin 3x)\right]}=\underset{x\to \frac{\pi }{2}}{lim}\frac{\sin x}{{\cos }^{-1}\left[{\sin }^{3}x\right]}$, $[\because \sin 3x=3\sin x-4{\sin }^{3}x]$

$L+=\underset{x\to \frac{\pi }{2}}{lim}\frac{\sin x}{{\cos }^{-1}0}=\frac{2}{\pi }$, $L-=\underset{x\to \frac{\pi }{2}}{lim}\frac{\sin x}{{\cos }^{-1}0}=\frac{2}{\pi }$

Hence required limit exist and is $=\frac{2}{\pi }$