## Problem Statement

Calculate the limit of the function:

$\lim _{x \rightarrow 0} \frac{\sqrt{1+x}-\sqrt{1-x}}{\sqrt[7]{x}}$
## Solution

$$
\begin{aligned}
& \lim _{x \rightarrow 0} \frac{\sqrt{1+x}-\sqrt{1-x}}{\sqrt[7]{x}}=\lim _{x \rightarrow 0} \frac{(\sqrt{1+x}-\sqrt{1-x})(\sqrt{1+x}+\sqrt{1-x})}{\sqrt[7]{x}(\sqrt{1+x}+\sqrt{1-x})}= \\
& =\lim _{x \rightarrow 0} \frac{1+x-(1-x)}{\sqrt[7]{x}(\sqrt{1+x}+\sqrt{1-x})}=\lim _{x \rightarrow 0} \frac{2 x}{x^{\frac{1}{7}}(\sqrt{1+x}+\sqrt{1-x})}= \\
& =\lim _{x \rightarrow 0} \frac{2 x^{\frac{6}{7}}}{\sqrt{1+x}+\sqrt{1-x}}=\frac{2 \cdot 0^{\frac{6}{7}}}{\sqrt{1+0}+\sqrt{1-0}}=\frac{0}{1+1}=0
\end{aligned}
$$

## Problem Kuznetsov Limits 11-22

This problem may have a different condition (possibly due to different editions or errors). See below for more details.