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  4. undersetx→ π /2 mathop lim ( sin x/ cos -1[ (1)4(3 sin x- sin 3x) ]), where [.] denotes greatest integer function, is

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

JamiaJamia 2013

Solution:

Given limit is $ L=\underset{x\to \frac{\pi }{2}}{\mathop{\lim }}\,\frac{\sin x}{{{\cos }^{-1}}[{{\sin }^{3}}x]} $ Now, when $ x\to \frac{\pi }{2},[{{\sin }^{3}}x]\to 0 $ as $ \sin x\to 1 $ $ \therefore $ $ L=\frac{2}{\pi } $