$l i m_{z \to 1} \frac{z ^{1 / 3} - 1}{z ^{1 / 6} - 1}$

Question

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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 $lim_{x \to a} \frac{x ^{n} - a ^{n}}{x - a} = n a^{n - 1}$

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

Answer 2

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