Question

Lim x³+1/x³-x quando x tende a -1

Answer 1

$L=\underset{x\to -1}{\mathrm{\ell im}}~\dfrac{x^3+1}{x^3-x}$


Temos uma indeterminação 0/0. Logo, podemos fatorar o numerador e o denominador por (x + 1):

$L=\underset{x\to -1}{\mathrm{\ell im}}~\dfrac{x^3+x^2-x^2+1}{x(x^2-1)}\\\\\\ =\underset{x\to -1}{\mathrm{\ell im}}~\dfrac{x^2(x+1)-x^2+1}{x(x+1)(x-1)}\\\\\\ =\underset{x\to -1}{\mathrm{\ell im}}~\dfrac{x^2(x+1)-(x^2-1)}{x(x+1)(x-1)}\\\\\\ =\underset{x\to -1}{\mathrm{\ell im}}~\dfrac{x^2(x+1)-(x-1)(x+1)}{x(x+1)(x-1)}\\\\\\ =\underset{x\to -1}{\mathrm{\ell im}}~\dfrac{\big(x^2-(x-1)\big)(x+1)}{x(x+1)(x-1)}\\\\\\ =\underset{x\to -1}{\mathrm{\ell im}}~\dfrac{(x^2-x+1)(x+1)}{x(x+1)(x-1)}$

$=\underset{x\to -1}{\mathrm{\ell im}}~\dfrac{x^2-x+1}{x(x-1)}\\\\\\ =\dfrac{(-1)^2-(-1)+1}{(-1)\big((-1)-1\big)}\\\\\\ =\dfrac{1+1+1}{(-1)\cdot (-2)}\\\\\\ =\dfrac{3}{2}~~~~~~\checkmark$


Dúvidas? Comente.


Bons estudos! :-)