Question

Lim (x²-4)/(√(x+2)-√(3x-2)), x-->2
Resolva fatorando

Answer 1

$\underset{x \to 2}{\mathrm{\ell im}}\;\dfrac{x^{2}-4}{\sqrt{x+2}-\sqrt{3x-2}}\\ \\ =\underset{x \to 2}{\mathrm{\ell im}}\;\dfrac{\left(x-2 \right )\cdot \left(x+2 \right )}{\sqrt{x+2}-\sqrt{3x-2}}$


Multiplicando o numerador e o denominador por $\left(\sqrt{x+2}+\sqrt{3x-2} \right )$, para eliminar as raízes do denominador:

$\underset{x \to 2}{\mathrm{\ell im}}\;\dfrac{\left(x-2 \right )\cdot \left(x+2 \right )\cdot \left(\sqrt{x+2}+\sqrt{3x-2} \right )}{\left(\sqrt{x+2}-\sqrt{3x-2} \right )\cdot \left(\sqrt{x+2}+\sqrt{3x-2} \right )}\\ \\ =\underset{x \to 2}{\mathrm{\ell im}}\;\dfrac{\left(x-2 \right )\cdot \left(x+2 \right )\cdot \left(\sqrt{x+2}+\sqrt{3x-2} \right )}{\left(\sqrt{x+2} \right )^{2}-\left(\sqrt{3x-2} \right )^{2}}\\ \\ =\underset{x \to 2}{\mathrm{\ell im}}\;\dfrac{\left(x-2 \right )\cdot \left(x+2 \right )\cdot \left(\sqrt{x+2}+\sqrt{3x-2} \right )}{\left(x+2 \right )-\left(3x-2 \right )}\\ \\ =\underset{x \to 2}{\mathrm{\ell im}}\;\dfrac{\left(x-2 \right )\cdot \left(x+2 \right )\cdot \left(\sqrt{x+2}+\sqrt{3x-2} \right )}{x+2-3x+2}\\ \\ =\underset{x \to 2}{\mathrm{\ell im}}\;\dfrac{\left(x-2 \right )\cdot \left(x+2 \right )\cdot \left(\sqrt{x+2}+\sqrt{3x-2} \right )}{-2x+4}$

$=\underset{x \to 2}{\mathrm{\ell im}}\;\dfrac{\left(x-2 \right )\cdot \left(x+2 \right )\cdot \left(\sqrt{x+2}+\sqrt{3x-2} \right )}{-2\cdot \left(x-2 \right )}$


Cancelando o fator $\left(x-2 \right )$, no numerador e no denominador, temos

$\underset{x \to 2}{\mathrm{\ell im}}\;\dfrac{\left(x+2 \right )\cdot \left(\sqrt{x+2}+\sqrt{3x-2} \right )}{-2}\\ \\ =\dfrac{\left(2+2 \right )\cdot \left(\sqrt{2+2}+\sqrt{3\cdot 2-2} \right )}{-2}\\ \\ =\dfrac{4\cdot \left(\sqrt{4}+\sqrt{6-2} \right )}{-2}\\ \\ =\dfrac{4\cdot \left(\sqrt{4}+\sqrt{4} \right )}{-2}\\ \\ =\dfrac{4\cdot \left(2+2 \right )}{-2}\\ \\ =\dfrac{4\cdot 4}{-2}\\ \\ =\dfrac{16}{-2}\\ \\ =-8\\ \\ \\ \boxed{ \begin{array}{c} \underset{x \to 2}{\mathrm{\ell im}}\;\dfrac{x^{2}-4}{\sqrt{x+2}-\sqrt{3x-2}}=-8 \end{array} }$