This limit does not exist. The limit from the right exists, but the left does not. There are no values for this function for x<2.

I think the square root was supposed to cover the entire problem.

`lim_(xrarr2) sqrt((x-2)/(x^2-4))`

`=lim_(xrarr2) sqrt((x-2)/(x^2-2^2))`

`=lim_(xrarr2) sqrt(((x-2))/((x+2)(x-2)))`

`=lim_(xrarr2) sqrt(1/(x+2))`

`= sqrt(1/(2+2))`

`= sqrt(1/4)`

`= 1/2`

*Therefore;*

`lim_(xrarr2) (sqrt((x-2)/(x^2-4))) = 1/2`