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Evaluate : $\lim\limits_{x\to\large\frac{\pi}{2}}\large\frac{\sin(\cos x)\cos x}{\sin x-cosec x}$

$(a)\;1\qquad(b)\;-1\qquad(c)\;0\qquad(d)\;2$

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$\lim\limits_{x\to\large\frac{\pi}{2}}\large\frac{\sin(\cos x)\cos x}{\sin x-cosec x}$$=\lim\limits_{x\to \Large\frac{\pi}{2}}\large\frac{\sin x(\cos x)\cos x}{\sin x-\large\frac{1}{\sin x}}$
$\Rightarrow \lim\limits_{x\to\large\frac{\pi}{2}}\large\frac{\sin(\cos x)\cos x\sin x}{-\cos^2x}$
$\Rightarrow -\lim\limits_{x\to\large\frac{\pi}{2}}\large\frac{\sin(\cos x)\sin x}{-\cos x}$
$\lim\limits_{x\to\large\frac{\pi}{2}}\large\frac{\sin(\cos x)}{\cos x}$$=1$
$\Rightarrow -1$
Hence (b) is the correct answer.
answered Dec 23, 2013 by sreemathi.v
 

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