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Define $f(0)$ so that $f(x)=(x+1)^{\cot x}$ becomes continuous at $x=0$

$\begin{array}{1 1}(A)\;f(0)=1 \\(B)\;f(0)=2 \\(C)\;f(0)=0 \\(D)\;f(0)=e \end{array}$

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$\lim \limits_{x \to 0} f(x) = \lim \limits_{x \to 0} (x+1)^{\cot x}$
$\qquad= \lim \limits_{x \to 0} \bigg[(1+x)^{\large\frac{1}{x}}\bigg]^{\large\frac{x}{\tan x}}$
$\qquad=e^1$
$\therefore f(0)=e$
Hence D is the correct answer.
answered Apr 22, 2014 by meena.p
 

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