Question

Evaluate the limit:
$\underset{x\to 0}{lim}\frac{\log (a+x)-\log a}{x}$

Answer 1

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

$=\underset{x\to 0}{lim}\frac{\log \left(\frac{a+x}{a}\right)}{a\times \frac{x}{a}}$
$[\because \log M-\log N=\log \frac{M}{N}]$

$=\underset{x\to 0}{lim}\frac{\log (1+\frac{x}{a})}{a\times \frac{x}{a}}$

$[\because \underset{x\to 0}{lim}\frac{\log (1+x)}{x}=1]$

$=\frac{1}{a}\times 1=\frac{1}{a}$

Therefore,
$\underset{x\to 0}{lim}\frac{\log (a+x)-\log a}{x}=\frac{1}{a}$