\(\displaystyle \lim_{x \rightarrow 0} \: \frac{\sqrt{1 + x}-\sqrt{1 - x}}{x}\)
\begin{equation*}
\begin{split}
& \lim_{x \rightarrow 0} \: \frac{\sqrt{1 + x}-\sqrt{1 - x}}{x} \\\\
& \lim_{x \rightarrow 0} \: \frac{\sqrt{1 + x}-\sqrt{1 - x}}{x} {\color {blue} \times \frac {\sqrt{1 + x} + \sqrt{1 - x}}{\sqrt{1 + x} + \sqrt{1 - x}}}\\\\
& \lim_{x \rightarrow 0} \: \frac{(1 + x) - (1 - x)}{x (\sqrt{1 + x} + \sqrt{1 - x})} \\\\
& \lim_{x \rightarrow 0} \: \frac{1 + x - 1 + x}{x (\sqrt{1 + x} + \sqrt{1 - x})} \\\\
& \lim_{x \rightarrow 0} \: \frac{2 \cancel{x}}{\cancel{x} (\sqrt{1 + x} + \sqrt{1 - x})} \\\\
& \lim_{x \rightarrow 0} \: \frac{2}{\sqrt{1 + x} + \sqrt{1 - x}} \\\\
& \frac{2}{\sqrt{1 + 0} + \sqrt{1 - 0}} \\\\
& \frac {2}{1 + 1} \\\\
& 1
\end{split}
\end{equation*}