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Question Number 101977 by Ar Brandon last updated on 05/Jul/20

$$\mathrm{Given}2\mathrm{functions},\mathrm{f}\mathrm{and}\mathrm{g},\mathrm{n}-\mathrm{times}\mathrm{derivable}\mathrm{within}$$$$\mathrm{the}\mathrm{open}\mathrm{interval},\mathbb{R}\mathrm{and}\mathrm{verify}\mathrm{the}\mathrm{property}$$$$\mathrm{f}\left({\mathrm{x}}_0\right)={\mathrm{f}}^{\left(\mathrm{k}\right)}\left({\mathrm{x}}_0\right)=0,\mathrm{g}\left({\mathrm{x}}_0\right)={\mathrm{g}}^{\left(\mathrm{k}\right)}\left({\mathrm{x}}_0\right)=0,\mathrm{\forall }\mathrm{k}\in \{1,2,...,\mathrm{n}-1\}$$$$\mathrm{Show}\mathrm{that}\underset{\mathrm{x}\to {\mathrm{x}}_0}{\lim }\frac{\mathrm{f}\left(\mathrm{x}\right)}{\mathrm{g}\left(\mathrm{x}\right)}=\frac{{\mathrm{f}}^{\left(\mathrm{n}\right)}\left({\mathrm{x}}_0\right)}{{\mathrm{g}}^{\left(\mathrm{n}\right)}\left({\mathrm{x}}_0\right)}$$

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