Question

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

Answer 1

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