Question

$\underset{x\to 2}{lim}\left(\frac{3^x+3^{3-x}-12}{3^{3-x}-3^{\frac{x}{2}}}\right)$

Answer 1

$\underset{x\to 2}{lim}\left(\frac{3^x+3^{3-x}-12}{3^{3-x}-3^{\frac{x}{2}}}\right)$ $\left(÷form\right)$

$L^{\prime }$ hospital Rule

$\underset{x\to 2}{lim}\frac{3^x\log 3+3^{3-x}\log \left(3\right)\left(-1\right)}{-3^{3-x}\log \left(3\right)-3^{\frac{x}{2}}\log 3\times \frac{1}{2}}$

$=\frac{3^2\log 3-3^{}\log 3}{-3\log 3-\frac{3}{2}\log 3}$

$=\frac{9\log 3-3log3}{\frac{-9}{2}\log 3}$

$=\frac{6\log 3}{\frac{-9}{2}\log 3}$

$=6\times \frac{-2}{9}$

$=\frac{-4}{3}$