Question

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

Answer 1

Answer: 3

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