If f(4)= 4, f'(4) =1 then `lim_(x to 4) 2((2-sqrtf(x))/ (2 - sqrtx))` is equal to

Options

A. `-1`
B. 1
C. 2
D. `-2`

Answer

B

Explanation

We have
`underset(x to 4)lim (2-sqrt(f(x)))/(2-sqrtx)`
`underset(x to 4)lim (4-f(x))/(4-x)xx (2+sqrtx)/(2+sqrt(f(x)))`
`underset(x to 4)lim (f(x)-4)/(x-4)xx underset(x to 4)lim (2+sqrtx)/(2+sqrt(f(x)))`
`=f'(4)xx (2+2)/(2+sqrt(f(4))) [{:(,therefore,"f(x) is differentiable at x=4"),(,therefore, "It is continuous at x=4"),(, ,"Hence "underset(x to 4)lim f(x)=f(4)):}]`
`=1xx(4)/(2+2)=1`