\newcommand{\combination}[2]{ \left( \begin{array}{c} #1 \\ #2 \end{array} \right ) }
\lim_{t \rightarrow 0} \frac{e^t -1}{t}
\leq \lim_{t \rightarrow 0} \left ( 1 + \frac{t}{n} \right ) ^{n+1} = \\
~ (*) ~\lim_{t \rightarrow 0} \frac{ 1 + \combination{n+1}{1}t/n + \combination{n+1}{2}(t/n)^2 + \cdots + \combination{n+1}{n+1}(t/n)^{n+1} - 1}{\displaystyle{t}}= \\=\lim_{t \rightarrow 0} \combination{n+1}{1}\frac{1}{n} + t*P(t) = \cdots