Question

Evaluate the Given limit: $\underset{z\to 1}{lim}\frac{z^{\frac{1}{3}}-1}{z^{\frac{1}{6}}-1}$

Answer 1

$\underset{z\to 1}{lim}\frac{z^{\frac{1}{3}}-1}{z^{\frac{1}{6}}-1}$
Put $z^{\frac{1}{6}}=x$
So,$z\to 1$ as $x\to 1$
$=\underset{x\to 1}{lim}\frac{x^2-1^2}{x-1}$
$=2\cdot 1^{2-1}$ $[\underset{x\to a}{lim}\frac{x^n-a^n}{x-a}=n{a}^{n-1}]$
$=2$
$\therefore \underset{z\to 1}{lim}\frac{z^{\frac{1}{3}}-1}{z^{\frac{1}{6}}-1}=2$