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$\lim\limits_{x\to 1}\big[\sec\big(\large\frac{\pi x}{2}\big)$$\log x\bigg]$ is

$(a)\;\large\frac{-2}{\pi}$$\qquad(b)\;-\large\frac{\pi}{2}$$\qquad(c)\;\large\frac{2}{\pi}$$\qquad(d)\;\large\frac{\pi}{2}$

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$\lim\limits_{x\to 1}\bigg[\sec\big(\large\frac{\pi x}{2}$$\big)\log x\bigg]$
$\Rightarrow \lim\limits_{x\to 1}\large\frac{\log x}{\cos\big(\Large\frac{\pi x}{2}\big)}\qquad\big(\large\frac{0}{0}\big)$ form
$\Rightarrow \lim\limits_{x\to 1}\large\frac{1/x}{-\cos(\Large\frac{\pi x}{2}\big).\Large\frac{\pi}{2}}$
$\Rightarrow -\large\frac{1}{\Large\frac{\pi}{2}\sin\frac{\pi}{2}}$
$\Rightarrow -\large\frac{2}{\pi}$
Hence (a) is the correct answer.
answered Jan 2, 2014 by sreemathi.v
 

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