If `f(x)={((sqrt(1+px)-sqrt(1-px))/(x) ",", -1 le x lt 0),((2x+1)/(x-2)",", 0 le x le 1):}` is continuous in [-1,1] then p is equal to

Options

A. 0
B. `-1`
C. `(1)/(2)`
D. `-(1)/(2)`

Answer

D

Explanation

Given, `f(x)={((sqrt(1+px)-sqrt(1-px))/(x)"," ,-1 le x lt 0),((2x+1)/(x-2)",",0le x le 1):}`
Since f(x) is continuous in [ -1, 1].
`therefore lim_(x to 0^(-)) f(x) = f(0)`
`rArr lim_(x to 0) (sqrt(1+px)-sqrt(1-px))/(x) xx (sqrt(1+px)+sqrt(1-px))/(sqrt(1+px)+sqrt(1-px))=(2(0)+1)/(0-2)` ltbtgt `rArr lim_(x to 0) (1+px-(1-px))/((sqrt(1+px)+sqrt(1-px)))=-(1)/(2)`
`rArr lim _(x to 0) (2p)/((sqrt(1+px)+sqrt(1-px)))=-(1)/(2)`
`rArr (2p)/(2) =-(1)/(2) rArr p = -(1)/(2) `