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If $f(x)=(x+1)^{\cot x}$ is continuous at $x=0$ then $f(0)$ is

$(a)\;0\qquad(b)\;\large\frac{1}{e}$$\qquad(c)\;e\qquad(d)\;None\;of\;these$

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$\lim\limits_{x\to 0}f(x)=\lim\limits_{x\to 0}(1+x)^{\Large\frac{1}{x}.\normalsize x\cot x}$
$\Rightarrow e^{\large\lim\limits_{x\to 0}x\cot x}$
$\Rightarrow e^{\large\lim\limits_{x\to 0}x\Large\frac{x}{\tan x}}$
$\Rightarrow e^1$
$\Rightarrow e$
Since $f(x)$ is continuous at $x=0$
$f(0)=e$
Hence (c) is the correct answer.
answered Dec 24, 2013 by sreemathi.v
 

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