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.

Hints:
$$e^x=\sum_{k=0}^\infty \frac{x^k}{k!}=1+x+\frac{x^2}{2!}+\ldots+\frac{x^6}{6!}+\frac{x^7}{7!}+\ldots\implies$$
$$e^x-1-x-\ldots-\frac{x^6}{6!}=\frac{x^7}{7!}+\frac{x^8}{8!}+O(x^9)\implies$$
$$\frac{e^x-1-x-\ldots-\frac{x^6}{6!}}{x^7}=\frac1{7!}+\frac{x}{8!}+O(x^2)\xrightarrow[x\to 0]{}\;?$$
