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$\lim _{x \rightarrow 0}\left(\frac{1}{x} \ln \sqrt{\frac{1+x}{1-x}}\right)$ is
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1
$\lim _{x \rightarrow 0} \frac{1 / 2(\ln (1+x)-\ln (1-x))}{x}$
$=\frac{1}{2} \lim _{x \rightarrow 0}\left(\frac{1}{1+x}+\frac{1}{1-x}\right)=1$
$=\frac{1}{2} \lim _{x \rightarrow 0}\left(\frac{1}{1+x}+\frac{1}{1-x}\right)=1$
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