Evaluate the given limit: mathop {lim }limits | vedclass

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(B) We are given the limit $\mathop {\lim }\limits_{z 	o 1} \frac{z^{1/3}-1}{z^{1/6}-1}$.At $z=1$,the expression takes the indeterminate form $\frac{

Vedclass · Accepted Answer

$2$

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(B) We are given the limit $\mathop {\lim }\limits_{z 	o 1} \frac{z^{1/3}-1}{z^{1/6}-1}$.At $z=1$,the expression takes the indeterminate form $\frac{0}{0}$.Let $z^{1/6} = x$. As $z 	o 1$,$x 	o 1$.Then $z^{1/3} = (z^{1/6})^2 = x^2$.The limit becomes $\mathop {\lim }\limits_{x 	o 1} \frac{x^2-1}{x-1}$.Using the standard formula $\mathop {\lim }\limits_{x 	o a} \frac{x^n-a^n}{x-a} = na^{n-1}$,where $n=2$ and $a=1$:$\mathop {\lim }\limits_{x 	o 1} \frac{x^2-1^2}{x-1} = 2(1)^{2-1} = 2(1) = 2$.Thus,the value of the limit is $2$.

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

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