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This is not strictly a question on differential equation but I think the topic asymptotic expansion is too closely related to D.E. and I think those experts in D.E. would help with my simple question here.

Please refer to the attached figure. I would like to know how one could come up with the asymptotic relations (1.89) to (1.91) for the function?

Is that we use the Taylor series to expand the function in power of epsilon with ingoring the exp(-x/epsilon) term completely?

And why an exp(-x/epsilon) term suddenly appear in relation (1.91)? This term is not come from any kind of expansion but just that the author wants to tell that exp(-x/epsilon) term is too small and it can be added anywhere without loss of the asymptoticity?

And when the domain becomes 0<=x<=2, ignoring exp(-x/epsilon) is no longer valid?

Are these the author want to tell?

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# An example regarding asymptotic expansion

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