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\displaystyle [br]\begin{equation*}\frac{e^x}{1-x} = \frac{1 + x + \frac{x^2}{2} + o(x^3)}{1-x} = \frac{\frac{1}{x} + 1 + \frac{x}{2} + o(x^2)}{\frac{1}{x} - 1}\end{equation*}
\displaystyle [br]\begin{equation*}\frac{e^x}{1-x} = \frac{1 + x + \frac{x^2}{2} + o(x^3)}{1-x} = \frac{\frac{1}{x} + 1 + \frac{x}{2} + o(x^2)}{\frac{1}{x} - 1}\end{equation*}
\displaystyle [br]\begin{equation*}\lim_{x \to \infty} \frac{e^x}{1-x} \stackrel{\text{DH}}{=} \lim_{x \to \infty} \frac{e^x}{-1} = -\infty\end{equation*}