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dimension10
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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.
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.
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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