Question

Limite de x quando tende a a

Answer 1

$\lim \limits_{x \to a}\cfrac{\sqrt[3]{x} - \sqrt[3]{a} }{x-a} \\\\= \lim \limits_{x \to a}\cfrac{\sqrt[3]{x} - \sqrt[3]{a} }{(\sqrt[3]{x})^3 - (\sqrt[3]{a})^3}$

Produto notável $a^3 - b^3 = (a-b)(a^2 + ab + b^2)$ :

$= \lim \limits_{x \to a}\cfrac{\sqrt[3]{x} - \sqrt[3]{a} }{(\sqrt[3]{x} - \sqrt[3]{a} )(\sqrt[3]{x^2} + \sqrt[3]{xa} + \sqrt[3]{a^2} )}\\\\= \lim \limits_{x \to a}\cfrac{1 }{\sqrt[3]{x^2} + \sqrt[3]{xa} + \sqrt[3]{a^2} }\\\\= \cfrac{1 }{\sqrt[3]{a^2} + \sqrt[3]{a^2} + \sqrt[3]{a^2} }\\\\= \cfrac{1 }{3\sqrt[3]{a^2}}$

$\boxed{\lim \limits_{x \to a}\cfrac{\sqrt[3]{x} - \sqrt[3]{a} }{x-a} = \cfrac{1 }{3\sqrt[3]{a^2}}}$