$\begin{aligned} & \lim _{x \rightarrow 2}(x-1)^{\frac{1}{3 x-6}} \\ & =\lim _{x \rightarrow 2}(x-2+1)^{\frac{1}{3(x-2)}}=\lim _{x \rightarrow 2}\left\{[1+(x-2)]^{\frac{1}{(x-2)}}\right\}^{\frac{1}{3}}=e^{\frac{1}{3}}\end{aligned}$