#lim_(x->2)((2-sqrt(8-x^2))/(x-2))#
#color(white)(888)#
#(2-sqrt(8-x^2))/(x-2)#
#color(white)(888)#
Multiply by #(2+sqrt(8-x^2))color(white)(88)# ( conjugate )
#color(white)(888)#
#((2+sqrt(8-x^2))(2-sqrt(8-x^2)))/((2+sqrt(8-x^2))(x-2))=(x^2-4)/((2+sqrt(8-x^2))(x-2)#
#color(white)(888)#
Factor numerator:
#((x+2)(x-2))/((2+sqrt(8-x^2))(x-2)#
Cancel:
#((x+2)cancel((x-2)))/((2+sqrt(8-x^2))cancel((x-2)))=((x+2))/((2+sqrt(8-x^2)))#
#color(white)(888)#
Plugging in #2#:
#color(white)(888)#
#((2+2))/((2+sqrt(8-(2)^2)))=4/4=1#
#:.#
#lim_(x->2)((2-sqrt(8-x^2))/(x-2))=1#