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Evaluate :$\lim\limits_{x\to 0}(\cos x)^{\large\frac{1}{x^2}}$

$(a)\;e\qquad(b)\;e^{\Large\frac{1}{2}}\qquad(c)\;e^{-\Large\frac{1}{2}}\qquad(d)\;0$

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Let $y=\lim\limits_{x\to 0}(\cos x)^{\large\frac{1}{x^2}}$
$\Rightarrow \lim\limits_{x\to 0}\large\frac{\log \cos x}{x^2}\qquad $$\big(0/0)$[By L Hospital rule]
$\Rightarrow \lim\limits_{x\to 0}\large\frac{1/cos x(-\sin x)}{2x}$
$\Rightarrow -\large\frac{1}{2}$$\lim\limits_{x\to 0}\large\frac{\tan x}{x}$
$\Rightarrow -\large\frac{1}{2}$
$y=e^{-\large\frac{1}{2}}$
Hence (c) is the correct answer.
answered Dec 23, 2013 by sreemathi.v
 

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