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Question Number 99262 by Ar Brandon last updated on 19/Jun/20

Consider f(x)=xe^(1/(1−x^2 ))   Find 3 real numbers a, b, and c such that  f(x) can be written in the form ax+b+(c/x)+(1/x)ξ(x)  where lim_(x→±∞) ξ(x)=0

$$\mathrm{Consider}\:\mathrm{f}\left(\mathrm{x}\right)=\mathrm{xe}^{\frac{\mathrm{1}}{\mathrm{1}−\mathrm{x}^{\mathrm{2}} }} \\ $$$$\mathrm{Find}\:\mathrm{3}\:\mathrm{real}\:\mathrm{numbers}\:\mathrm{a},\:\mathrm{b},\:\mathrm{and}\:\mathrm{c}\:\mathrm{such}\:\mathrm{that} \\ $$$$\mathrm{f}\left(\mathrm{x}\right)\:\mathrm{can}\:\mathrm{be}\:\mathrm{written}\:\mathrm{in}\:\mathrm{the}\:\mathrm{form}\:\mathrm{ax}+\mathrm{b}+\frac{\mathrm{c}}{\mathrm{x}}+\frac{\mathrm{1}}{\mathrm{x}}\xi\left(\mathrm{x}\right) \\ $$$$\mathrm{where}\:\underset{\mathrm{x}\rightarrow\pm\infty} {\mathrm{lim}}\xi\left(\mathrm{x}\right)=\mathrm{0} \\ $$

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