Question

The value of $\displaystyle\lim _{x \rightarrow 0} \frac{\cos (\sin x)-\cos x}{x^{4}}$ is equal to

• A. $1 / 5$
• B. $1 / 6$
• C. $1 / 4$
• D. $1 / 2$

Answer 1

Answer: (B)

$\frac{2 \cos \left(\frac{\sin x+x}{2}\right) \cdot \sin \left(\frac{x-\sin x}{2}\right)}{x^{4}}$
$=\displaystyle\lim _{x \rightarrow 0} 2\left[\frac{\sin \left(\frac{\sin x+x}{2}\right)}{\left(\frac{\sin x+ x}{2}\right)}\right]\left[\frac{\sin \left(\frac{x-\sin x}{2}\right)}{\left(\frac{x-\sin x}{2}\right)}\right]$
$\times\left[\frac{1}{2\left(\frac{1}{\frac{\sin x}{x}+1}\right) \cdot 3 \frac{x^{3}}{(x-\sin x)}}\right]$
$=\frac{1}{6}$