Answer
$$\frac{2}{3}$$
Work Step by Step
\begin{align*}
\lim _{x \rightarrow 1} \frac{x^{1 / 3}-1}{\sqrt{x}-1}&=\lim _{x \rightarrow 1} \frac{\left(x^{1 / 3}-1\right)}{(\sqrt{x}-1)} \cdot \frac{\left(x^{2 / 3}+x^{1 / 3}+1\right)(\sqrt{x}+1)}{(\sqrt{x}+1)\left(x^{2 / 3}+x^{1 / 3}+1\right)}\\
&=\lim _{x \rightarrow 1} \frac{(x-1)(\sqrt{x+1})}{(x-1)\left(x^{2 / 3}+x^{1 / 3}+1\right)}\\
&=\lim _{x \rightarrow 1} \frac{\sqrt{x}+1}{x^{2 / 3}+x^{1 / 3}+1}\\
&=\frac{1+1}{1+1+1}\\
&=\frac{2}{3}
\end{align*}
