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Question-113530




Question Number 113530 by Khalmohmmad last updated on 13/Sep/20
Answered by Aziztisffola last updated on 13/Sep/20
−∞×(+∞)=−∞
$$−\infty×\left(+\infty\right)=−\infty \\ $$
Answered by mathmax by abdo last updated on 13/Sep/20
let f(x) =((lnx)/x)  changement x =(1/t) give f(x)=f((1/t))=−tln(t)  x→0^+  ⇒t→+∞ ⇒lim_(t→+∞) −tlnt =−∞ ⇒lim_(x→0^+ )   ((lnx)/x)=−∞
$$\mathrm{let}\:\mathrm{f}\left(\mathrm{x}\right)\:=\frac{\mathrm{lnx}}{\mathrm{x}}\:\:\mathrm{changement}\:\mathrm{x}\:=\frac{\mathrm{1}}{\mathrm{t}}\:\mathrm{give}\:\mathrm{f}\left(\mathrm{x}\right)=\mathrm{f}\left(\frac{\mathrm{1}}{\mathrm{t}}\right)=−\mathrm{tln}\left(\mathrm{t}\right) \\ $$$$\mathrm{x}\rightarrow\mathrm{0}^{+} \:\Rightarrow\mathrm{t}\rightarrow+\infty\:\Rightarrow\mathrm{lim}_{\mathrm{t}\rightarrow+\infty} −\mathrm{tlnt}\:=−\infty\:\Rightarrow\mathrm{lim}_{\mathrm{x}\rightarrow\mathrm{0}^{+} } \:\:\frac{\mathrm{lnx}}{\mathrm{x}}=−\infty \\ $$

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