Question

$\underset{x\to 2}{lim}\frac{x-2}{x^{1/3}-2^{1/3}}$

Answer 1

${lim}_{x\to 2}\frac{x-2}{x^{1/3}-2^{1/3}}={lim}_{x\to 2}\left(\frac{x-2}{\sqrt[3]{3x}-\sqrt[3]{32}}\right)$

Using L'Hospital rule,

${lim}_{x\to 2}\left(\frac{1}{1/3x^{2/3}}\right)$

${lim}_{x\to 2}\left(3x^{2/3}\right)$

$={3.2}^{2/3}$

$\approx 4.76$