`lim_(h->0) (f(2h+2+h^2)-f(2))/(f(h-h^2+1)-f(1))` given that `f'(2)=6 and f'(1)=4` then (a) limit does not exist (b) is equal to `- 3/2` (c) is equal to `3/2` (d) is equal to 3

Options

A. does not exist
B. is equal to `-(3)/(2)`
C. is equal to `(3)/(2)`
D. is equal to 3

Answer

D

Explanation

We have
`underset(h to 0)lim (f(2h+2+h^(2))-f(2))/(f(h-h^(2)+1)-f(1))`
`=underset(h to 0)lim (f(2h+2+h^(2))-f(2))/((2h+2+h^(2))-2)xx((2h+2+h^(2))-2)/((h-h^(2)+1)-1)xx(1)/((f(h-h^(2)+1)-f(1))/((h-h^(2)+1)-1))`
`=f'(2)xxunderset(h to 0)lim (2+h)/(1-h)xx(1)/(f'(1))`
`=(2f'(2))/(f'(1))=(12)/(4)=3`