How to solvelim_(x->2)(2-sqrt(8-x^2))/(x-2) ?

1 Answer
Dec 22, 2017

1

Explanation:

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