$\underset{x\to 2}{lim} \frac{3^x+3^{x-1}-12}{3^{-x/2}-3^{1-x}}$ is equal to

$\underset{x\to 2}{lim} \frac{3^x+\frac{3^x}{3}-12}{3^{-x/2}-\frac{3}{3^x}}=\underset{x\to 2}{lim} \frac{3^x\left(4\cdot 3^x-36\right)}{3\left(3^{x/2}-3\right)}$

$\begin{array}{l}=\underset{x\to 2}{lim} \frac{4\times 3^x\left(3^x-3^2\right)}{3\left(3^{x/2}-3\right)}=\underset{x\to 2}{lim} \frac{4\times 3^{x-1}\left(3^{x/2}-3\right)\left(3^{x/2}+3\right)}{\left(3^{x/2}-3\right)}\\ =4\times 3(3+3)=72\end{array}$