<s>[INST] seeming ugly limit i want to compute the limit
$$\lim_{x \rightarrow 0} \frac{e^x-1-x-\frac{x^2}{2}-\frac{x^3}{6}-\frac{x^4}{24}-\frac{x^5}{120}-\frac{x^6}{720}}{x^7}$$
Instead of doing some messy calculation, I think if there is some ingenious way to compute this limit, but i don't know how to do. thank you so much.
 [/INST] Apply the L'Hôpital rule $7$ times (until the derivatives kill the polynomial $1+x+\ldots+\frac{x^6}{6!}$) and we find
$$\lim_{x \rightarrow 0} \frac{e^x-1-x-\frac{x^2}{2}-\frac{x^3}{6}-\frac{x^4}{24}-\frac{x^5}{120}-\frac{x^6}{720}}{x^7}=\lim_{x \rightarrow 0}\frac{e^x}{7!}=\frac{1}{7!}$$
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