lim_(x->2)((2-sqrt(8-x^2))/(x-2))
color(white)(888)
(2-sqrt(8-x^2))/(x-2)
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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