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  4. The value of displaystyle lim x arrow 0 (x3 cot x/1- cos x) is

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Q. The value of $\displaystyle\lim _{x \rightarrow 0} \frac{x^{3} \cot x}{1-\cos x}$ is

Limits and Derivatives

Solution:

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