Given that `lim_(x to oo ) ((2+x^(2))/(1+x)-Ax-B)=3`
If `G(x)=sqrt(25-x^(2))` then what is `lim_(xto1) (G(x)-G(1))/(x-1)` equal to?

Options

A. `-(sqrt1)/(2sqrt6)`
B. `(1)/(5)`
C. `-(1)/(sqrt6)`
D. `(1)/(sqrt6)`

Answer

A

Explanation

`G(x)=sqrt(25-x)`
Now,
`underset(xto1)lim(G(x)-G(1))/(x-1)`
`underset(xto1)lim(sqrt(25-x^(2))-sqrt(24))/(x-1)`
Applying L' Hospital rule,
`underset(xto1)lim(-2x)/(2sqrt(25-x^(2)))=(-2)/(2sqrt(25-x^(2)))`
`=(-2)/(2sqrt24)=(-1)/(sqrt24)=(-1)/(2sqrt6)`
`:.` Option (a) is correct.