<s>[INST] Finding the error in an evaluation of the limit $\lim_{x\to0} \frac{e^x-x-1}{x^2} $ \begin{align}
\lim_{x\to0} \frac{e^x-x-1}{x^2}
&= \lim_{x\to0} \frac{e^x-1}{x^2} - \lim_{x\to0} \frac{1}{x} \\
&= \lim_{x\to0} \frac{e^x-1}{x}\lim_{x\to0} \frac{1}{x} - \lim_{x\to0} \frac{1}{x} \\ &= \lim_{x\to0} \frac{1}{x} - \lim_{x\to0} \frac{1}{x} \\
&= 0
\end{align}
 [/INST] The first claimed equality,
$$\lim_{x\to0} \frac{e^x-x-1}{x^2} \color{red}{\stackrel{\textrm{(false)}}{=}} \lim_{x\to0} \frac{e^x-1}{x^2} - \lim_{x\to0} \frac{1}{x} ,$$ already doesn't hold: Neither of the limits on its r.h.s. exist.
To evaluate the limit:
Hint Apply l'Hopital's Rule (twice) or expand $e^x$ in a Maclaurin series.
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