Question #061a3

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Answer 1 · 2017-12-27T00:04:44

$-5/6$

$lim_(xrarr2)((sqrt(x^2+5)-4x+5)/(x^2-4))$

$lim_(xrarr2)(((sqrt(x^2+5)-(4x-5))(sqrt(x^2+5)+(4x-5)))/((x^2-4)(sqrt(x^2+5)+(4x-5))))$ (conjugate both sides)

$lim_(xrarr2)((x^2+5-(4x-5)^2)/((x^2-4)(sqrt(x^2+5)+(4x-5))))$

$lim_(xrarr2)((x^2+5-16x^2+40x-25)/((x^2-4)(sqrt(x^2+5)+(4x-5))))$

$lim_(xrarr2)((-15x^2+40x-20)/((x^2-4)(sqrt(x^2+5)+(4x-5))))$

$lim_(xrarr2)((-5(3x^2-8x+4))/((x^2-4)(sqrt(x^2+5)+(4x-5))))$

$lim_(xrarr2)((-5(3x-2)(x-2))/((x+2)(x-2)(sqrt(x^2+5)+(4x-5))))$

$lim_(xrarr2)((-5(3x-2))/((x+2)(sqrt(x^2+5)+(4x-5))))$ (factor out x-2)

direct substitution: $=(-5(3(2)-2))/((2+2)(sqrt(2^2+5)+(4(2)-5)))$

$=(-5(4))/(4(sqrt(9)+3))$

$=-20/(4(6))$

$=-5/6$