Question

Evaluate the limit:
$\underset{x\to 0}{lim}\frac{5^x+3^x+2^x-3}{x}$

Answer 1

Given:
$\underset{x\to 0}{lim}\frac{5^x+3^x+2^x-3}{x}$

$=\underset{x\to 0}{lim}\frac{5^x-1+3^x-1+2^x-1}{x}$

$=\underset{x\to 0}{lim}(\frac{5^x-1}{x}+\frac{3^x-1}{x}+\frac{2^x-1}{x})$

$=\underset{x\to 0}{lim}\left(\frac{5^x-1}{x}\right)+\underset{x\to 0}{lim}\left(\frac{3^x-1}{x}\right)+\underset{x\to 0}{lim}\left(\frac{2^x-1}{x}\right)$

$=\log 5+\log 3+\log 2[\underset{x\to 0}{lim}\left(\frac{a^x-1}{x}\right)=\log a]$

$=\log (5\cdot 3\cdot 2)[\because \log M+\log N=\log MN]$
$=\log 30$