Question

If $L =\displaystyle\lim _{ x \rightarrow 0} \frac{ e ^{- x ^{2} / 2}-\cos x }{ x ^{3} \sin x }$ then the value of $1 /(3 L )$ is

Answer 1

$\displaystyle\lim _{x \rightarrow 0} \frac{e^{-x^{2} / 2}-\cos x}{x^{3} \sin x}$
$=\displaystyle\lim _{x \rightarrow 0} \frac{\left(1-\frac{\left(x^{2} / 2\right)}{1 !}+\frac{\left(x^{2} / 2\right)^{2}}{2 !}-\left(1-\frac{x^{2}}{2 !}+\frac{x^{4}}{4 !}\right)\right)}{x^{3}\left(x-\frac{x^{3}}{3 !}\right)}$
$=\displaystyle\lim _{x \rightarrow 0} \frac{\left(\frac{x^{4}}{8}\right)-\left(\frac{x^{4}}{24}\right)}{x^{4}\left(1-\frac{x^{2}}{3 !}\right)}=\frac{1}{12}$