How do you evaluate the limit (sqrt(x^2-5)+2)/(x-3)x25+2x3 as x approaches 33?

1 Answer
Aug 14, 2018

Please go through the Discussion in The Explanation.

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.