Find the values of a and b in order that
`lim_(xto0) (x(1+acosx)-bsinx)/(x^(3))=1.`

Answer

`a=(-5)/(2),b=(-3)/(2)`

Explanation

`underset(xto0)lim(x(1+acosx)-bsinx)/(x^(3))=1" "`(0/0 form)
`impliesunderset(xto0)lim(x+axcosx-bsinx)/(x^(3))=1`
`impliesunderset(xto0)lim(x+ax(1-(x^(2))/(2!)+(x^(4))/(4!)+...)-b(x-(x^(3))/(3!)+(x^(5))/(5!)+...))/(x^(3))=1`
`impliesunderset(xto0)lim((1+a-b)x+(-(a)/(2)+(b)/(6))x^(3))/(x^(3))=1`
`implies1+a-b=0" "(1)`
and `-a/2+b/6=1" "(2)`
Solving equations (1) and (2), we get
`a=-(5)/(2),b=-(3)/(2).`