I was wondering, what would be the solution for [tex]\lim_{x \rightarrow -\infty} W(x) [/tex] where W(x) is the Lambert-W function.(adsbygoogle = window.adsbygoogle || []).push({});

Wolfram|Alpha gives the result [tex]\infty[/tex] but the graph certainly does not imply this (In fact,[tex]\lim_{x\rightarrow \infty} x\,\exp(x)=\infty [/tex]) I only graphed it it between -10 and 10 and behind -1.5 or so, the graph goes straight down so you can't even see the function beyond -1.75 or so. Does the function make a sharp turn somewhere else behind? Is the function even continuous?

Also,

[tex]\lim_{\omega \rightarrow -\infty} \omega \,\exp(\omega)=0[/tex] and [tex]\lim_{\omega\rightarrow 0} \omega\,\exp(\omega)=0[/tex]. So does this imply that [tex] W(0)=0 \mbox{ and } W(-\infty)=0 [/tex].

And one final question, Wolfram|Alpha gives a long complicated result for the inverse function of x^x involving the Lambert W function. How does one use the Lambert W function to do so?

Thanks.

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# Some questions about the Lambert W function

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