Question

limit x tends to 0 3^2x-2^3x/x​

Answer 1

$\sf{\displaystyle\lim _{x\to \:0}\left(\dfrac{3^{2x}-2^{3x}}{x}\right)}$

$\sf{\displaystyle=\lim _{x\to \:0}\left(\frac{9^x-8^x}{x}\right)}$

Applying L'Hopital's Rule

$\sf{\displaystyle=\lim _{x\to \:0}\left(\frac{2\ln \left(3\right)\cdot \:9^x-3\ln \left(2\right)\cdot \:8^x}{1}\right)}$

$\sf{\displaystyle=\lim _{x\to \:0}\left(2\ln \left(3\right)\cdot \:9^x-3\ln \left(2\right)\cdot \:8^x\right)}$

$\sf{\displaystyle=2\ln \left(3\right)\cdot \:9^0-3\ln \left(2\right)\cdot \:8^0}$

$\sf{\displaystyle=2\ln \left(3\right)-3\ln \left(2\right)}$