Evaluate the following limit: _ x 0 (3+x)- (3-x) x - Vidyadip

Question

Evaluate the following limit:
$\lim\limits_{\text{x}\rightarrow0}\frac{\log\text{(3}+\text{x})-\log\text{(3}-\text{x})}{\text{x}}$

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Answer

$\lim\limits_{\text{x}\rightarrow0}\frac{\log\text{(3}+\text{x})-\log\text{(3}-\text{x})}{\text{x}}$
$=\lim\limits_{\text{x}\rightarrow0}\frac{\text{log}\Big(\frac{3+\text{x}}{3-\text{x}}\Big)}{\text{x}}$
$=\lim\limits_{\text{x}\rightarrow0}\frac{\text{log}\Big(1+\frac{2\text{x}}{3-\text{x}}\Big)}{\text{x}}$
$=\lim\limits_{\text{x}\rightarrow0}\frac{\text{log}\Big(1+\frac{2\text{x}}{3-\text{x}}\Big)}{\frac{2\text{x}}{3-\text{x}}}\times\lim\limits_{\text{x}\rightarrow0}\frac{2}{3-\text{x}}$
$=\frac{2}{3}$

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