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The value of $\lim _{x \rightarrow 0} \frac{x^{3} \cot x}{1-\cos x}$ is
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$\lim _{x \rightarrow 0} \frac{x^{3} \cot x}{1-\cos x}$
$=\lim _{x \rightarrow 0}\left(\frac{x^{3} \cot x}{1-\cos x} \times \frac{1+\cos x}{1+\cos x}\right)$
$=\operatorname{x}_{x \rightarrow 0}\left(\frac{3}{\sin x}\right) \times \lim _{x \rightarrow 0} \cos x \times \lim _{x \rightarrow 0}(1+\cos x)=2$
$=\lim _{x \rightarrow 0}\left(\frac{x^{3} \cot x}{1-\cos x} \times \frac{1+\cos x}{1+\cos x}\right)$
$=\operatorname{x}_{x \rightarrow 0}\left(\frac{3}{\sin x}\right) \times \lim _{x \rightarrow 0} \cos x \times \lim _{x \rightarrow 0}(1+\cos x)=2$
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