calculus - $lim_{xto0}frac{e^x-1-x}{x^2}$ using only rules of algebra of limits.

I would like to solve that limit solved using only rules of algebra of limits.

$$\lim_{x\to0}\frac{e^x-1-x}{x^2}$$

All the answers in How to find $\lim\limits_{x\to0}\frac{e^x-1-x}{x^2}$ without using l'Hopital's rule nor any series expansion? do not fully address my question.

A challenging limit problem for the level of student who knows that:

$$\begin{align*} \lim\limits_{x\to +\infty} e^x&=+\infty\tag1\\ \lim\limits_{x\to -\infty} e^x&=0\tag2\\ \lim\limits_{x\to +\infty} \frac{e^x}{x^n}&=+\infty\tag3\\ \lim\limits_{x\to -\infty} x^ne^x&=0\tag4\\ \lim\limits_{x\to 0} \frac{e^x-1}{x}&=1\tag5 \end{align*}$$