The value of `lim_(xto2) (sqrt(1+sqrt(2+x))-sqrt(3))/(x-2)" is "`

Options

A. 0
B. `e^(x)`
C. `log_(e)x`
D. none of these

Answer

A

Explanation

`underset(xto2)lim(sqrt(1+sqrt(2+x))-sqrt(3))/(x-2)`
`underset(xto2)lim(1+sqrt(2+x)-3)/((sqrt(1+sqrt(2+x))+sqrt(3))(x-2))" "`(Rationalizing)
`=underset(xto2)"lim"(sqrt(2+x)-2)/((sqrt(1+sqrt(2+x))+sqrt(3)")"(x-2)))`
`=underset(xto2)"lim"((x-2))/("("sqrt(1+sqrt(2+x))+sqrt(3)")""("sqrt(2+x)+2")""("x-2")")`
(Rationalizing)
`=(1)/("("2sqrt(3")")4)=(1)/(8sqrt(3))`