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Evaluate $\lim \limits_{x \to 0} \bigg(\large\frac{x^3 \cot x}{1- \cos x} \bigg)$

$\begin{array}{1 1}(A)\;4 \\(B)\;2 \\(C)\;0 \\(D)\;1 \end{array}$

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$\lim \limits_{x \to 0} \bigg( \large \frac{x^3 \cot x}{1- \cot x}\bigg)$
$\qquad= \lim _{x \to 0} \bigg \{\large\frac{x^3 \cot x }{(1-\cos x)} \times \large\frac{(1+ \cos x )}{(1+\cos x)} \bigg\}$
$\qquad= \lim _{x \to 0} \large\frac{x^3 \cos x (1+\cos x)}{\sin x(1-\cos ^2 x)}$
$\qquad= \lim _{x \to 0} \large\frac{x^3 \cos x (1+\cos x)}{\sin ^3 x}$
$\qquad= \lim _{x \to 0} \bigg( \large\frac{x}{\sin x } \bigg)^3$$ \times \lim \limits _{x \to 0} \cos x \times \lim\limits _{x \to 0} (1+\cos x)$
$\qquad= (1^3 \times 1 \times 2)$
$\qquad=2$
Hence B is the correct answer
answered Apr 22, 2014 by meena.p
 

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