Question

Evaluate the Given limit: $\lim_{z\rightarrow 1}$ $\frac{z^{\frac{1}{3}}-1}{z^{\frac{1}{6}}-1}$

Answer 1

At z = 1, the value of the given function takes the form 0/0.
Put so that z →1 as x →1.
$\lim_{z\rightarrow 1}$ $\frac{z^{\frac{1}{3}}-1}{z^{\frac{1}{6}}-1}$
Accordingly \lim_{z\rightarrow 1}$$\frac{z^{\frac{1}{3}}-1}{z^{\frac{1}{6}}-1} = $\lim_{x\rightarrow 1}$ $\frac{x^2-1}{x-1}$
=$\lim_{x\rightarrow 1}$ $\frac{x^2-1}{x-1}$
=2.1 2-1 [$\lim_{x\rightarrow a}$ $\frac{x^n-a^n}{x-a}$ = nan-1]
=2
∴$\lim_{z\rightarrow 1}$ $\frac{z^{\frac{1}{3}}-1}{z^{\frac{1}{6}}-1}$ = 2