Question

Lim X___> 2

X - 2
_______
∛3X - 5 -1

(A raíz é cúbica se não tiver dando pra ver)

Answer 1

$\mathrm{\lim\limits_{x\to2}\dfrac{x-2}{\sqrt[3]{\mathrm{3x-5}}-1}=\dfrac{2-2}{\sqrt[3]{3.2-5}-1}=\dfrac{0}{\sqrt[3]{1}-1}=\dfrac{0}{0}}\\\\\\ \textbf{Aplicando a Regra de L'Hopital:}\\\\\ \mathrm{\lim\limits_{x\to2}=\dfrac{(x-2)'}{((3x-5)^{\frac{1}{3}}-1)'}=\lim\limits_{x\to2}\dfrac{1}{\frac{1}{3}(3x-5)^{-\frac{2}{3}}.3}=}\\\\ \mathrm{=\lim\limits_{x\to2}\dfrac{1}{\dfrac{1}{(3x-5)^{\frac{2}{3}}}}=\lim\limits_{x\to2}(3x-5)^{\frac{2}{3}}=\lim\limits_{x\to2}\sqrt[3]{\mathrm{(3x-5)^2}}=}\\\\ \mathrm{=\sqrt[3]{(3.2-5)^2}=\sqrt[3]{(6-5)^2}=\sqrt[3]{(1)^2}=\boxed{\mathbf{1}}}$