Evaluate `lim_(xto0)(tanx-sinx)/(x^(3)).`

Explanation

We have `underset(xto0)lim(tanx-sinx)/(x^(3))" "`(0/0 form)
`=underset(xto0)lim((sinx-sinxcosx)/(x^(3)cosx))`
`=underset(xto0)lim{(sinx(1-cosx))/(x^(3)cosx)}=underset(xto0)lim{(sinx)/(x)(1-cosx)/(x^(2))(1)/(cosx)}`
`={underset(xto0)lim(sinx)/(x)}{underset(xto0)lim(2"sin"^(2)(x)/(2))/(((x)/(2))xx4)}underset(xto0)lim(1)/(cosx)`
`={underset(xto0)lim(sinx)/(x)}1/2{underset(xto0)lim((sin(x/2))/((x)/2))^(2)}underset(xto0)lim(1)/(cosx)`
`=1xx1/2(1)^(2)xx1/2=1/2`