If `f(a)=(1)/(4)`, then `lim_(hrarr0) (f(a+2h^(2))-f(a-2h^(2)))/(f(a+h^(3)-h^(2))-f(a-h^(3)+h^(2)))`=

Options

A. 0
B. 1
C. `-2`
D. none of these

Answer

C

Explanation

`underset(hrarr0)(lim)(f(a+2h^(2))-f(a-2h^(2)))/(f(a+h^(3)-h^(2))-f(a-h^(3)+h^(2)))`
`=underset(hrarr0)(lim)(f'(a+2h^(2))4h-f'(1-2h^(2))(-4h))/(f'(a+h^(3)-h^(2))(3h^(2)-2h)-f'(a-h^(3)+h^(2))(-3h^(2)+2h))`
`=underset(hrarr0)(lim)(4h)/(3h^(2)-2h)[(f'(a+2h^(2))+f'(a-2h^(2)))/(f'(a+h^(3)-h^(2))+f'(a-h^(3)+h^(2)))]`
`=underset(hrarr0)(lim)(4)/(3h-2)=-2`