x\log x

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4.23 Limits of $x\log x$

4.23.1 Example

The following equivalent statements hold:

(i) $e^{u}/u\rightarrow\infty$ as $u\rightarrow\infty$ ;

(ii) $ue^{-u}\rightarrow 0$ as $u\rightarrow\infty$ ;

(iii) $x\log x\rightarrow 0$ as $x\rightarrow 0+$ .

Solution. (i) For $u>0$ , we have

e^{u}=1+u+{{u^{2}}\over{2!}}+\dots\geq{{u^{2}}\over{2}},

so $e^{u}/u\geq u/2$ ; hence $e^{u}/u\rightarrow\infty$ as $u\rightarrow\infty$ .