The value of lim _{x rightarrow 0} left( frac | vedclass

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(C) Let $L = \lim _{x \rightarrow 0} \frac{1}{x} \ln \sqrt{\frac{1+x}{1-x}}$.Using the property $\ln(a^b) = b \ln(a)$,we can rewrite the expression as

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(C) Let $L = \lim _{x \rightarrow 0} \frac{1}{x} \ln \sqrt{\frac{1+x}{1-x}}$.Using the property $\ln(a^b) = b \ln(a)$,we can rewrite the expression as:$L = \lim _{x \rightarrow 0} \frac{1}{x} \cdot \frac{1}{2} \ln \left( \frac{1+x}{1-x} \right)$.$L = \frac{1}{2} \lim _{x \rightarrow 0} \frac{\ln(1+x) - \ln(1-x)}{x}$.Using the standard limit $\lim _{x \rightarrow 0} \frac{\ln(1+ax)}{x} = a$,we have:$L = \frac{1}{2} \left( \lim _{x \rightarrow 0} \frac{\ln(1+x)}{x} - \lim _{x \rightarrow 0} \frac{\ln(1-x)}{x} \right)$.$L = \frac{1}{2} (1 - (-1)) = \frac{1}{2} (2) = 1$.

The value of $\lim _{x \rightarrow 0} \left( \frac{1}{x} \ln \sqrt{\frac{1+x}{1-x}} \right)$ is

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