Evaluate lim(x→0)[(3^2x-2^3x)/x]

Question

Evaluate: 

\(\lim\limits_{x \to 0}(\frac{3^{2x}-2^{3x}}{x})\)

Answer 1

\(\lim\limits_{x \to 0}(\frac{3^{2x}-2^{3x}}{x})\) 

\(=\lim\limits_{x \to 0}\{\frac{(3^{2x}-1)-(2^{3x}-1)}{x}\}\) 

\(=\lim\limits_{x \to 0}\frac{3^{2x}-1}{x}-\lim\limits_{x \to 0}\frac{2^{3x}-1}{x}\) 

\(=\lim\limits_{x \to 0}\frac{1}{2}[\frac{3^{2x}-1}{x}]-3.[\lim\limits_{x \to 0}\frac{2^{3x}-1}{x}]\) 

= 2 ∙ (log 3) − 3(log 2)

= log3² − log2³ 

= log \(\frac{9}{8}\)