
\begin{itemize}
\item $\limite[x<1]{x}{1}{\left(2\e^{x-1}\right)}=2\times\e^0=2>0$
\item $\limite[x<1]{x}{1}{\big(\e^{x-1}-1\big)}=0^-$
\end{itemize}
Ainsi, par quotient des limites,
\[
\boxed{ \limite[x<1]{x}{1}{\left(\dfrac{2\e^{x-1}}{\e^{x-1}-1}\right)} = -\infty }
\]
