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Let $f(x)=\left\{\begin{array}{1 1}\big[\tan(\large\frac{\pi}{4}\normalsize +x)\big]^{\large\frac{1}{x}}&x\neq 0\\k&x=0\end{array}\right.$.For what value of k,$f(x)$ is continuous at $x=0$

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

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$\lim\limits_{x\to 0}f(x)=\lim\limits_{x\to 0}\big[\tan(\large\frac{\pi}{4}$$+x\big)\big]^{1/x}$
$\Rightarrow \lim\limits_{x\to 0}\big(\large\frac{1+\tan x}{1-\tan x}\big)^{1/x}$
$\Rightarrow \lim\limits_{x\to 0}\big[(1+\tan x)^{1/\tan x}\big]^{\large\frac{\tan x}{x}}\times \lim\limits_{x\to 0}\big[(1-\tan x)^{-1/\tan x}\big]^{\large\frac{\tan x}{x}}$
$\Rightarrow e.e$
$\Rightarrow e^2$
Hence (d) is the correct answer.
answered Dec 30, 2013 by sreemathi.v
 

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