## Problem Statement

Calculate the limit of the function:

$\lim _{x \rightarrow 1} \frac{\sqrt{x-1}}{\sqrt[3]{x^{2}-1}}$
## Solution

$$
\begin{aligned}
& \lim _{x \rightarrow 1} \frac{\sqrt{x-1}}{\sqrt[3]{x^{2}-1}}=\left\{\frac{0}{0}\right\}=\lim _{x \rightarrow 1} \frac{\sqrt{x-1} \sqrt{x+1}}{\sqrt[3]{x^{2}-1} \sqrt{x+1}}= \\
& =\lim _{x \rightarrow 1} \frac{\sqrt{x^{2}-1}}{\sqrt[3]{x^{2}-1} \sqrt{x+1}}=\lim _{x \rightarrow 1} \frac{\sqrt[6]{\left(x^{2}-1\right)^{3}}}{\sqrt[6]{\left(x^{2}-1\right)^{2}} \sqrt{x+1}}= \\
& =\lim _{x \rightarrow 1} \frac{\sqrt[6]{x^{2}-1}}{\sqrt{x+1}}=\frac{\sqrt[6]{1^{2}-1}}{\sqrt{1+1}}=\frac{0}{2}=0
\end{aligned}
$$

## Problem Kuznetsov Limits 11-3