Flat function/Polynomial in 1 over x times exp -1 over x/Limit for x to 0/Exercise

We consider the function

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Prove that for all ${}n\in \mathbb {N}$ the ${}n$-th derivative ${}f^{(n)}$ satisfies the following property

${}\operatorname {lim} _{x\rightarrow 0}\,f^{(n)}(x)=0\,.$

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