If `f(x)=sqet(25-x^(2))`, then what is `Lim_(x to 1) (f(x)-f(1))/(x-1)` equal to?

Options

A. `(1)/(5)`
B. `(1)/(24)`
C. `sqrt24`
D. `-(sqrt1)/(sqrt24)`

Answer

D

Explanation

`f(x)=sqrt(25-x^(2))`
`f(1)=sqrt(24)`
`impliesunderset(xto1)lim(f(x)-f(1))/(x-1)`
`:."It is "(0)/(0)` (undefined conditions) so using L' hospital's rule
`impliesunderset(xto1)lim(f'(x)-0)/(1)=underset(xto1)lim(sqrt(25-x^(2)))`
`impliesunderset(xto12)lim(1)/(2)xx(1)/(sqrt(25-(1)^(2)))(-2x)`
`=(1)/(2)xx(1)/(sqrt(25-(1)^(2)))xx(-2)`
`=-(1)/(sqrt(24))`