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

[haywood]

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

 

[dalle]

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.

 

[haywood]

That works! I was especially looking for how to find that second root, so grateful that you mentioned using ln :-)
Thank you, Dalle!