Question

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

Answer 1

$$\begin{gathered} \underset{x\to 0}{\lim }\left[\frac{\log \left(a+x\right)-\log \left(a-x\right)}{x}\right] \\ =\underset{x\to 0}{\lim }\left[\frac{\log \left({\displaystyle \frac{a+x}{a-x}}\right)}{x}\right] \\ =\underset{x\to 0}{\lim }\left[\frac{\log \left(1+{\displaystyle \frac{a+x}{a-x}}-1\right)}{x}\right] \\ =\underset{x\to 0}{\lim }\left[\frac{\log \left(1+{\displaystyle \frac{a+x-a+x}{a-x}}\right)}{x}\right] \\ =\underset{x\to 0}{\lim }\left[\frac{\log \left(1+{\displaystyle \frac{2x}{a-x}}\right)}{{\displaystyle \frac{2x}{a-x}}\times \left({\displaystyle \frac{a-x}{2}}\right)}\right] \\ x\to 0 \\ \therefore \frac{2x}{a-x}\to 0 \\ \mathrm{Let}y=\frac{2x}{a-x} \\ =\underset{y\to 0}{\lim }\left[\frac{\log \left(1+y\right)}{y}\right]\times \underset{x\to 0}{\lim }\left(\frac{1}{{\displaystyle \frac{a-x}{2}}}\right) \\ =1\times \frac{2}{a} \end{gathered}$$