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Question Number 85985 by Rio Michael last updated on 26/Mar/20

Consider the functionf defined by parf(x) = −x + ((ln x)/x) in the interval  : ]0,+∞[.  (C_f ) is its representative curve in an orthonormal  reference system (O,i^→ ,j^→ ).   Calculate  lim_(x→0^+  )  f(x), lim_(x→+∞)  f(x).

$$\mathrm{Consider}\:\mathrm{the}\:\mathrm{function}{f}\:\mathrm{defined}\:\mathrm{by}\:\mathrm{par}{f}\left({x}\right)\:=\:−{x}\:+\:\frac{\mathrm{ln}\:{x}}{{x}}\:\mathrm{in}\:\mathrm{the}\:\mathrm{interval} \\ $$$$\left.:\:\right]\mathrm{0},+\infty\left[.\:\:\left({C}_{{f}} \right)\:\mathrm{is}\:\mathrm{its}\:\mathrm{representative}\:\mathrm{curve}\:\mathrm{in}\:\mathrm{an}\:\mathrm{orthonormal}\right. \\ $$$$\mathrm{reference}\:\mathrm{system}\:\left(\mathrm{O},\overset{\rightarrow} {{i}},\overset{\rightarrow} {{j}}\right). \\ $$$$\:\mathrm{Calculate}\:\:\underset{{x}\rightarrow\mathrm{0}^{+} \:} {\mathrm{lim}}\:{f}\left({x}\right),\:\underset{{x}\rightarrow+\infty} {\mathrm{lim}}\:{f}\left({x}\right). \\ $$

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