If `f^(prime)(a)=1/4, t h e n(lim)_(hvec0)(f(a+2h^2)-f(a-2h^2))/(f(a+h^3-h^2)-f(a-h^3+h^2))=` `0` b. `1` c. `-2` d. none of these

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`