\begin{array}{l}
& =\lim \_{x \rightarrow 0}\left(\frac{e^{x}}{n} \frac{e^{n x}-1}{e^{x}-1}\right)^{\frac{e}{x}} \\
&=\operatorname{exp}\left[\lim \_{x \rightarrow 0} \frac{e}{x} \ln \left(\frac{e^{x}}{n} \frac{e^{n x}-1}{e^{x}-1}\right) \right] \\
&=\exp \left[e \lim \_{x \rightarrow 0} \frac{\ln \left(\frac{e^{x}}{n} \frac{e^{nx}-1}{e^{x}-1}\right)}{x}\right] \\
&\text { As } \quad \lim \_{x \rightarrow 0} \ln \left(\frac{e^{x}}{n} \frac{e^{n x}-1}{e^{x}-1}\right)=\lim \_{x \rightarrow 0}\left(\frac{1}{n} \frac{n e^{n x}}{e^{x}}\right) 
=1 \\
&\text { But } \quad \lim \_{x \rightarrow 0} x=0
\end{array}

So the result is infinity. What am I doing wrong?