Question

If `L=lim_(xto0) (e^(-x^(2)//2)-cosx)/(x^(3)sinx)`, then the value of `1//(3L)` is ________.

Answer 1

Correct Answer - `(4)`
`underset(xto0)lim(e^(-x^(2)//2)-cosx)/(x^(3)sinx)`
`=underset(xto0)lim((1-((x^(2)//2))/(1!)+((x^(2)//2)^(2))/(2!))-(1-x^(2)/(2!)+x^(4)/(4!)))/(x^(3)(x-(x^(3))/(3!)))`
`=underset(xto0)lim(((x^(4))/(8))-((x^(4))/(24)))/(x^(4)(1-(x^(2))/(3!)))=(1)/(12)`