Question

Lim [√(x ² - 2x + 6) - √(x ² + 2x - 6)] / (x ² - 4x + 3) =
x-> 3

Answer 1

Resposta:

-1/3

Explicação passo-a-passo:

Temos:

lim [√(x²-2x+6) - √(x²+2x-6)]/(x²-4x+3)

x->3

lim [√(x²-2x+6)-√(x²+2x-6)]/(x²-4x+3) .

x->3 [√(x²-2x+6)+√(x²+2x-6)]/

[√(x²-2x+6)+√(x²+2x-6)]

lim [{√(x²-2x+6)}²-{√(x²+2x-6)}²] /

x->3 {(x²-4x+3).[√(x²-2x+6)+√(x²+2x-6)]}

lim [x² -2x +6 -x² -2x +6] / {(x²-4x+3).

x->3 [√(x²-2x+6)+√(x²+2x-6)]}

lim (-4x +12) / {(x²-4x+3).[√(x²-2x+6)+

x->3 √(x²+2x-6)]}

Temos, aplicando o método de Briott-Ruffini, que:

x² -4x +3 | -4x +12

-------------

-x² +3x -(1/4)x +(1/4)

-----------

0 -x +3

x -3

-------

0

Logo:

lim {(-4x +12)/(x²-4x+3)} . {1/

x->3 {[√(x²-2x+6) + √(x²+2x-6)]}

lim {1/[-(1/4)x +(1/4)]} . {1/

x->3 {[√(x²-2x+6) + √(x²+2x-6)]}

lim {1/[(-x+1)/4]} . {1/

x->3 {[√(x²-2x+6) + √(x²+2x-6)]}

lim {4/(1-x)} . {1/{[√(x²-2x+6) +

x->3 √(x²+2x-6)]}

Substituindo x para o valor tendendo a 3, temos:

{4/(1-3)} . {1/{[√(3²-2.3+6) + √(3²+2.3-6)]}

{4/(-2)} . {1/{[√(9-6+6) + √(9+6-6)]}

(-2) . 1/(√9 + √9)

-2/(3+3)

-2/6

-1/3

Blz?

Abs :)