Let `f ''(x)` be continuous at `x=0` If `("lim")_(xvec0)(2f(x)-3a(f(2x)+bf(8x))/(sin^2x)` exists and `f(0)!=0,f^(prime)(0)!=0,` then the value of `(3a)/b` is____

Answer

`(7)`

Explanation

We have
`L=underset(xto0)lim(2f(x)-3af(2x)+bf(8x))/(sin^(2)x)`
For the limit to exist, we have
`2f(0)-3af(0)+bf(0)=0`
or `3a-b =2" "[becausef(0)ne0," given"]" "(1)`
or `L=underset(xto0)lim(2f'(x)-6af'(2x)+8bf'(8x))/(2x)`
For the limit to exist, we have
`2f'(0)-6af'(0)+8bf'(0)=0`
`implies" "3a-4b=1" "[becousef'(0)ne0," given"]" "(2)`
Solving equations (1) and (2), we have `a=7//9` and `b=1//3`.