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Question Number 4118 by Filup last updated on 29/Dec/15
Does a function f(x) exist  such that for  f^( (n)) (x)=(d^n f/dx^n )  That:  (1)     lim_(n→k)  f^( (n)) (x)=k  and  (2)     lim_(n→k)  f^( (n)) (x)=f(k)
$$\mathrm{Does}\:\mathrm{a}\:\mathrm{function}\:{f}\left({x}\right)\:\mathrm{exist} \\ $$$$\mathrm{such}\:\mathrm{that}\:\mathrm{for} \\ $$$${f}^{\:\left({n}\right)} \left({x}\right)=\frac{{d}^{{n}} {f}}{{dx}^{{n}} } \\ $$$$\mathrm{That}: \\ $$$$\left(\mathrm{1}\right)\:\:\:\:\:\underset{{n}\rightarrow{k}} {\mathrm{lim}}\:{f}^{\:\left({n}\right)} \left({x}\right)={k} \\ $$$$\boldsymbol{\mathrm{and}} \\ $$$$\left(\mathrm{2}\right)\:\:\:\:\:\underset{{n}\rightarrow{k}} {\mathrm{lim}}\:{f}^{\:\left({n}\right)} \left({x}\right)={f}\left({k}\right) \\ $$
Commented by prakash jain last updated on 29/Dec/15
lim_(n→k)  f^((n)) (x)=k=f(k)?
$$\underset{{n}\rightarrow{k}} {\mathrm{lim}}\:{f}^{\left({n}\right)} \left({x}\right)={k}={f}\left({k}\right)? \\ $$
Commented by prakash jain last updated on 29/Dec/15
Either f(k)=k or limit does not exist.
$$\mathrm{Either}\:{f}\left({k}\right)={k}\:\mathrm{or}\:\mathrm{limit}\:\mathrm{does}\:\mathrm{not}\:\mathrm{exist}. \\ $$

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