Question

Evaluate the limit:
$\underset{x\to 0}{lim}\left(\frac{a^x-a^{-x}}{x}\right)$

Answer 1

Given :
$\underset{x\to 0}{lim}\left(\frac{a^x-a^{-x}}{x}\right)$

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

$=\underset{x\to 0}{lim}\left(\frac{a^{2x}-1}{x{a}^x}\right)$

$=\underset{x\to 0}{lim}\left(\frac{a^{2x}-1}{2x}\right)\times \frac{2}{a^x}$

$=\underset{x\to 0}{lim}\left(\frac{a^{2x}-1}{x{a}^x}\right)\times \underset{x\to 0}{lim}\frac{2}{a^x}$

$=\log a\times \frac{2}{a^0}$

$=\log a\times \frac{2}{1}[\because {lim}_{x\to 0}\left(\frac{a^x-1}{x}\right)=\log a]$

$=2\log a$

Therefore,
$\underset{x\to 0}{lim}\left(\frac{a^x-a^{-x}}{x}\right)=2\log a$