Question

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

Answer 1

Answer: 4

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