How do you evaluate the limit #(sqrt(x^2-5)+2)/(x-3)# as x approaches #3#?

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Answer 1 · 2018-08-14T05:47:28

Please go through the Discussion in The Explanation.

Observe that,

#lim_(x to 3)(sqrt(x^2-5)+2)=sqrt(3^2-5)+2=2+2=4#.

Hence, as long as #(x-3)# remains in the Denominator , the

limit can not exist.

#"However, had the Required Limit been "lim_(x to 3){sqrt(x^2-5)-2}/(x-3)#,

#"The Limit"=lim{sqrt(x^2-5)-2}/(x-3)xx{sqrt(x^2-5)+2}/{sqrt(x^2-5)+2}#,

#=lim{(sqrt(x^2-5))^2-2^2}/[(x-3){sqrt(x^2-5)+2}]#,

#=lim(x^2-5-4)/[(x-3){sqrt(x^2-5)+2}]#,

#=lim_(x to 3){cancel((x-3))(x+3)}/[cancel((x-3)){sqrt(x^2-5)+2}]#,

#=(3+3)/{sqrt(3^2-5)+2}#,

#=6/(2+2)#,

#=3/2#.