Evaluate _ z 1 z^ 1/3 - 1 z^ 1/6 - 1 - Vidyadip

Question

Evaluate $\mathop {\lim }\limits_{z \to 1} \frac{{{z^{1/3}} - 1}}{{{z^{1/6}} - 1}}$

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Answer

Here $\mathop {\lim }\limits_{z \to 1} \frac{{{z^{1/3}} - 1}}{{{z^{1/6}} - 1}}\left[ {\frac{0}{0}{\text{form}}} \right]$
$= \mathop {\lim }\limits_{z \to 1} \frac{{{{({z^{1/6}} - 1)}^2}}}{{{z^{1/6}} - 1}}$
$ = \mathop {\lim }\limits_{z \to 1} \frac{{({z^{1/6}} + 1)({z^{1/6}} - 1)}}{{({z^{1/6}} - 1)}} = \mathop {\lim }\limits_{z \to 1}$
$= z^{1/6} + 1$
$= (1)^{1/6} + 1 = 1 + 1 = 2$

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