$\underset{x→0}{\lim}\begin{pmatrix}\frac{2^x+3^x}{2}\end{pmatrix}^{2/x}$ is equal to

6

$\underset{x→0}{\lim}\begin{pmatrix}\frac{2^x+3^x}{2}\end{pmatrix}^{2/x}$  [1^∞ form]

$e^{\underset{x→0}{\lim}\frac{2}{x}\begin{pmatrix}\frac{2^x+3^x}{2}-1\end{pmatrix}}=e^{\underset{x→0}{\lim}\begin{pmatrix}\frac{2^x-1}{x}+\frac{3^x-1}{x}\end{pmatrix}}=e^{\log 2+\log 3}=6$