Question

lim x->3 ∛x - ∛3 / x-3

Answer 1

$\lim_{x \to 3} \frac{ \sqrt[3]{x}- \sqrt[3]{3} }{x-3}$

diferença de dois cubos
a³-b³ = (a-b)(a²+ab+b²)

aplicando isso para eliminar as raízes no numerador e ficar (x-3)
temos a=∛x , b=∛3
temos que multiplicar por (a²+ab+b²)
então multiplicando o numerador e o denominador

$\lim_{x \to 3} \frac{ \sqrt[3]{x}- \sqrt[3]{3} }{x-3} * \frac{[ (\sqrt[3]{x})^2- \sqrt[3]{x}\sqrt[3]{3}+(\sqrt[3]{3})^2] }{[ (\sqrt[3]{x})^2- \sqrt[3]{x}\sqrt[3]{3}+(\sqrt[3]{3})^2] } \\\\ \lim_{x \to 3} \frac{( \sqrt[3]{x})^3-( \sqrt[3]{3} )^3 }{(x-3)*[(\sqrt[3]{x})^2- \sqrt[3]{x}\sqrt[3]{3}+(\sqrt[3]{3})^2] } \\\\ \lim_{x \to 3} \frac{(x-3) }{(x-3)*[(\sqrt[3]{x})^2- \sqrt[3]{x}\sqrt[3]{3}+(\sqrt[3]{3})^2] } \\\\ \lim_{x \to 3} \frac{1}{(\sqrt[3]{x})^2- \sqrt[3]{x}\sqrt[3]{3}+(\sqrt[3]{3})^2}$

$= \frac{1}{(\sqrt[3]{3})^2+ \sqrt[3]{3}\sqrt[3]{3}+(\sqrt[3]{3})^2} = \frac{1}{3(\sqrt[3]{3})^2} = \frac{1}{3 \sqrt[3]{9} } = \frac{\sqrt[3]{3} }{9}$