Find the value of `lim_(xto0) (sinx+log_(e)(sqrt(1+sin^(2)x)-sinx))/(sin^(3)x).`

Explanation

Let `sin x=t.` Therefore,
`underset(xto0)lim(sinx+log_(e)(sqrt(1+sin^(2)x)-sinx))/(sin^(3)x)`
`=underset(t to0)lim(t+log_(e)(sqrt(1+t^(2))-1))/(t^(3))`
`=underset(t to0)lim(1+(1)/(sqrt(1+t^(2))-t)((t)/(sqrt(1+t^(2)))-1))/(3t^(2))`
(Using L'Hospital's Rule)
`=underset(t to0)lim(1-(1)/(sqrt(1+t^(2))))/(3t^(2))=underset(t to0)lim(sqrt(1+t^(2))-1)/(3t^(2)sqrt(1+t^(2)))`
`=underset(t to0)lim((1+t^(2))-1)/(3t^(2))xx(1)/((sqrt(1+t^(2))+1))=1/6`