How to solve#lim_(x->2)(2-sqrt(8-x^2))/(x-2)# ?

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Answer 1 · 2017-12-22T13:22:53

#1#

#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)#

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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)))#

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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#