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

Question

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

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Answer

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

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