Question

${lim}_{x\to 0}\frac{\log (a+x)-\log (a-x)}{x},a>0$

Answer 1

$={lim}_{x\to 0}\left(\frac{log\left(\frac{a+x}{a-x}\right)}{x}\right)$

$={lim}_{x\to 0}\left(\frac{log(1+(\frac{2x}{a-x}\left)\right)}{\left(\frac{2x}{a-x}\right)\times x}\right)\times \left(\frac{2x}{(a-x)}\right)$

$use{lim}_{x\to 0}\left(\frac{log(1+x)}{x}\right)=1$

$\therefore limitvalue=1\times \left(\frac{2}{a-0}\right)=\left(\frac{2}{a}\right)$