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 [\(\lim_{x\rightarrow a}\) \(\frac{x^n-a^n}{x-a}\) = na^n-1]
=2
∴\(\lim_{z\rightarrow 1}\) \(\frac{z^{\frac{1}{3}}-1}{z^{\frac{1}{6}}-1}\) = 2