Finding Two-Term Asymptotic Expansion for Real Roots of xe-x=epsilon

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To find a two-term asymptotic expansion for the real roots of the equation xe^(-x) = epsilon as epsilon approaches 0, a Taylor polynomial can be used for the root near 0. For the second root, taking logarithms of both sides is an effective approach. This method provides a clearer pathway to approximate the roots. The discussion highlights the importance of these techniques in deriving asymptotic expansions. Overall, the conversation focuses on practical methods for solving the equation.
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Can anyone tell me how to find a two-term asymptotic expansion for the two real roots of xe-x=epsilon as epsilon --> 0

Thanks,
A.Haywood
 
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dear Haywood
I don't know what kind of asypmtotic expasion you are looking for, so a short hint on how to solve the equation approximatly must suffice.
For the root near 0 substitute the left hand side with a taylor polynomial.
For the other root take logarithms on both sides.
 
That works! I was especially looking for how to find that second root, so grateful that you mentioned using ln :-)
Thank you, Dalle!
 
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