Question

$li{m}_{z\to 1}\frac{z^{1/3}-1}{z^{1/6}-1}$

Answer 1

${lim}_{z\to 1}\frac{z^{\frac{1}{3}}-1}{z^{\frac{1}{6}}-1}$

$={lim}_{z\to 1}\frac{\frac{z^{\frac{1}{3}}-1}{z-1}}{\frac{z^{\frac{1}{6}}-1}{z-1}}$

$=\frac{\frac{1}{3}\times 1^{\frac{-2}{3}}}{\frac{1}{6}\times 1^{\frac{-5}{6}}}$ since ${lim}_{x\to a}\frac{x^n-a^n}{x-a}=n{a}^{n-1}$

$=\frac{6}{3}=2$