Question

Solve $\underset{x\to 0}{lim}\frac{3^{2x}-1}{2^{3x}-1}$

Answer 1

$\underset{x\to 0}{lim}\frac{3^{2x}-1}{2^{3x}-1}$
$=\underset{x\to 0}{lim}\frac{(9^x-1).x}{(9^n-1).x}$
$=\underset{x\to 0}{lim}\frac{\left(\frac{9^n-1}{x}\right)}{\left(\frac{8^n-1}{x}\right)}$
$=\frac{\underset{x\to 0}{lim}\left(\frac{9^n-1}{x}\right)}{\underset{x\to 0}{lim}\left(\frac{8^n-1}{x}\right)}=\frac{ln9}{ln8}$
$=lo{g}_8^9$
$=lo{g}_{2^3}^{3^2}$
$=\frac{2}{3}lo{g}_2^3$