Question

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

Answer 1

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

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

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

$=\frac{1}{a}$