[tex]e^x-1=x+\frac{x^2}{2}+\frac{x^3}{6}+\mathcal{O}\left ( x^4 \right )\Rightarrow \lim\limits_{x\to 0}\frac{e^x-1-x}{x^2}=\lim\limits_{x\to 0}\frac{x+\frac{x^2}{2}+\frac{x^3}{6}+\mathcal{O}\left ( x^4 \right )-x}{x^2}=\\=\lim\limits_{x\to 0}\frac{1}{x^2}\left ( \frac{x^2}{2}+\frac{x^3}{6}+\mathcal{O}\left ( x^4 \right ) \right )=\lim\limits_{x\to 0}\left ( \frac{1}{2}+\frac{x}{6}+\ldots \right )=\frac{1}{2}[/tex]