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Find $\lim\limits_{x\to 0}\{\tan(\large\frac{\pi}{4}$$+x)\}^{1/x}$

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

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$\lim\limits_{x\to 0}\{\tan(\large\frac{\pi}{4})$$+x\}^{1/x}=e^{\large\lim\limits_{x\to 0}\log\{\tan(\Large\frac{\pi}{4}+x)\}^{1/x}}$
Using $\lim\limits_{x\to 0}f(x)=e^{\large\lim\limits_{x\to 0}\log f(x)}$
$\Rightarrow e^{\large\lim\limits_{x\to 0}\Large\frac{\log \tan((\pi/4)+x)}{x}}\qquad[0/0]$ form
Using L Hospital's rule
$\Rightarrow e^{\Large\lim\limits_{x\to 0}\bigg[\Large\frac{\sec^2(\pi/4+x)}{\tan(\pi/4)+x}\bigg]}$
$\Rightarrow e^{\Large\frac{2}{1}}=e^2$
Hence (c) is the correct answer.
answered Dec 31, 2013 by sreemathi.v
 

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