_ x 0 x^3 x 1 - x = - Vidyadip

Question

$\mathop {\lim }\limits_{x \to 0} \frac{{{x^3}\cot x}}{{1 - \cos x}} = $

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Answer

c
(c) $\mathop {\lim }\limits_{x \to 0} \,\,\frac{{{x^3}\cot x}}{{1 - \cos x}} = \mathop {\lim }\limits_{x \to 0} \,\left( {\frac{{{x^3}\cot x}}{{1 - \cos x}} \times \frac{{1 + \cos x}}{{1 + \cos x}}} \right)$

$ = \mathop {\lim }\limits_{x \to 0} \,{\left( {\frac{x}{{\sin x}}} \right)^3} \times \mathop {\lim }\limits_{x \to 0} \,\cos x \times \mathop {\lim }\limits_{x \to 0} \,(1 + \cos x) = 2$

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