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Q. The value of $ \lim \limits_{x\to0} \frac{cos \left(sin\, x\right) - cos\,x}{x^{4}} $

MHT CETMHT CET 2007

Solution:

$\displaystyle\lim _{x \rightarrow 0} \frac{\cos (\sin x)-\cos x}{x^{4}}$
$=\displaystyle\lim _{x \rightarrow 0} \frac{2 \sin \left(\frac{x+\sin x}{2}\right) \sin \left(\frac{x-\sin x}{2}\right)}{x^{4}}$
$=2 \displaystyle\lim _{x \rightarrow 0}\left[\frac{\sin \left(\frac{x+\sin x}{2}\right)}{\left(\frac{x+\sin x}{2}\right)} \times \frac{\sin \left(\frac{x-\sin x}{2}\right)}{\left(\frac{x-\sin x}{2}\right)}\right.$
$\left.\times\left(\frac{x+\sin x}{2 x}\right)\left(\frac{x-\sin x}{2 x^{3}}\right)\right]$
$=2 \displaystyle\lim _{x \rightarrow 0}\left[\frac{\sin \left(\frac{x+\sin x}{2}\right)}{\frac{x+\sin x}{2}} \times \frac{\sin \left(\frac{x-\sin x}{2}\right)}{\frac{x-\sin x}{2}}\right.$
$\times\left(\frac{1}{2}+\frac{\sin x}{2 x}\right)\left(\frac{x-\sin x}{2 x^{3}}\right)\bigg]$
$=2 \times 1 \times 1 \times\left(\frac{1}{2}+\frac{1}{2}\right) \lim _{x \rightarrow 0} \frac{x-\sin x}{2 x^{3}}$
$=\displaystyle\lim _{x \rightarrow 0} \frac{x-\sin x}{x^{3}}=\displaystyle\lim _{x \rightarrow 0} \frac{x-\left(x-\frac{x^{3}}{3 !}+\frac{x^{5}}{5 !}-\ldots\right)}{x^{3}}$
$=\displaystyle\lim _{x \rightarrow 0}\left(\frac{1}{3 !}-\frac{x^{2}}{5 !}+\ldots\right)=\frac{1}{6}$