$\begin{aligned} \lim _{x \rightarrow 1}(\log e x)^{1 / \log x} &=\lim _{x \rightarrow 1}[\log e+\log x]^{1 / \log x} \\ &=\lim _{x \rightarrow 1}[1+\log x]^{1 / \log x} \\ &=e^{\lim _{x \rightarrow 1} \frac{\log x}{\log x}}=e \end{aligned}$