If `("lim")_(x -> 0)(cos4x+acos2x+b)/(x^4)` is finite, find `aa n db` using expansion formula.

Explanation

`underset(xto0)lim(cos4x+acos2x+b)/(x^(4))="Finite"`
Using expansion formula for `cos4x" and "cos2x,` we get
`underset(xto0)lim((1-(4x)^(2)/(2!)+(4x)^(4)/(4!))+a(1-(2x)^(2)/(2!)+(2x)^(4)/(4!))+b)/(x^(4))`
or `" "underset(xto0)lim((1+a+b)+(-8-2a)x^(2)+((32)/(3)+2/3a)x^(4)+...)/(x^(4))`
or `" "1+a+b=0`
`-8-2=0`
Solving (1) and (2) for a and b, we get
`a=-4" and " b=3`
Also, Limit`=32/3+2/3a=(32-8)/(3)=8`