Question

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

Answer 1

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

$=\underset{x\to 0}{lim}\frac{a^x+b^x-2-c^x-d^x+2}{x}$

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

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

$=\log a+\log b-\log c-\log d$

$=(\log a+\log b)-(\log c+\log d)$

$=\log \left(ab\right)-\log \left(cd\right)$
$[\because \log M+\log N=\log MN]$

$=\log \frac{ab}{cd}$
$[\because \log M-\log N=\log \frac{M}{N}]$

Therefore,
$\underset{x\to 0}{lim}\frac{a^x+b^x-c^x-d^x}{x}=\log \frac{ab}{cd}$