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  4. The value of displaystyle lim x arrow 0 (( sin x-x)2+(1- cos x3)/x5 sin x) is

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Q. The value of $\displaystyle\lim _{x \rightarrow 0} \frac{(\sin x-x)^{2}+\left(1-\cos x^{3}\right)}{x^{5} \sin x}$ is

Limits and Derivatives

Solution:

$\displaystyle\lim _{x \rightarrow 0} \frac{(\sin x-x)^{2}+\left(1-\cos x^{3}\right)}{x^{5} \sin x}$
$=\displaystyle\lim _{x \rightarrow 0} \frac{(\sin x-x)^{2}+\left(1-\cos x^{3}\right)}{x^{6} \frac{\sin x}{x}}$
$=\displaystyle\lim _{x \rightarrow 0}\left(\frac{\sin x-x}{x^{3}}\right)^{2}+\left(\frac{1-\cos x^{3}}{x^{6}}\right)$
$=\displaystyle\lim _{x \rightarrow 0}\left(\frac{x-\frac{x^{3}}{3 !}-x}{x^{3}}\right)^{2}+\left(\frac{1-\left(1-\frac{x^{6}}{2 !}\right)}{x^{6}}\right)$
$=\left(\frac{-1}{6}\right)^{2}+\frac{1}{2}=\frac{19}{36}$