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  4. The value of displaystyle lim x arrow 0 (2(1-( cos )3 x)/( sin )2 x cos x) is equal to

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Q. The value of $\displaystyle\lim _{x \rightarrow 0} \frac{2\left(1-(\cos )^3 x\right)}{(\sin )^2 x \cos x}$ is equal to

NTA AbhyasNTA Abhyas 2022

Solution:

$\displaystyle\lim _{x \rightarrow 0} \frac{2\left(1-(\cos )^3 x\right)}{(\sin )^2 x \cos x}$
$=\displaystyle\lim _{x \rightarrow 0} \frac{2(1-\cos x)\left(1+(\cos )^2 x+\cos x\right)}{(\sin )^2 x \cos x}$
$=\displaystyle\lim _{x \rightarrow 0} \frac{4(\sin )^2 \frac{x}{2}\left(1+(\cos )^2 x+\cos x\right)}{\left(\left\{2 \sin \frac{x}{2} \cos \frac{x}{2}\right\}\right)^2 \cos x}$
$=\displaystyle\lim _{x \rightarrow 0} \frac{1+\cos ^2 x+\cos x}{\left\{\cos \frac{x}{2}\right\}^2 \cos x}$
$=3$