Question

Evaluate the limit:
$\underset{x\to 0}{lim}\frac{a^x+b^x-2}{x}$

Answer 1

Given:
$\underset{x\to 0}{lim}\frac{a^x+b^x-2}{x}$

$=\underset{x\to 0}{lim}\frac{a^x-1+b^x-1}{x}$

$=\underset{x\to 0}{lim}(\frac{a^x-1}{x}+\frac{b^x-1}{x})$

$=\underset{x\to 0}{lim}\left(\frac{a^x-1}{x}\right)+\underset{x\to 0}{lim}\left(\frac{b^x-1}{x}\right)$

$=\log a+\log b[\because {lim}_{x\to 0}\left(\frac{a^x-1}{x}\right)=\log a]$

$=\log ab[\because \log M+\log N=\log MN]$

$\therefore \underset{x\to 0}{lim}\frac{a^x+b^x-2}{x}=\log ab$