# Asymptotic Expansion around minus infinity

Consider

 f[x_, a_] := Simplify[Sqrt[a*x + x^2], Assumptions -> x \[Element] Reals]


Then if I run:

 Series[f[x, 0], {x, Infinity, 1}]

Series[f[x, 1], {x, Infinity, 1}]


The first results in Abs[x] as expected but the leading term of the second Series is x. How do I force it to be Abs[x]? The background is that this is a simplified version of a more complicated function and I am interested in the asymptotic behavior at both +Infinity and -Infinity.

• Welcome to stackexchange, you can format code with the {} button, or starting a new paragraph with four spaces. – Feyre Aug 31 '16 at 14:50
• It is not clear, why are you thinking about the absolute value? You have in mind a real x and expand it in the vicinity of the infinity. It is positive. The result of your first expression Series[f1[x, 0], {x, Infinity, 1}] was for me (MMa11, Win7) as follows: SeriesData[x, DirectedInfinity[1], {1}, -1, 2, 1], Not Abs[x]. – Alexei Boulbitch Aug 31 '16 at 14:56
• I would also like to know the asymptotic behavior at -Infinity. Running – user42740 Aug 31 '16 at 14:58
• The expansion at -infinity does show a long-standing problem with Series. About all I can say is I am hoping to see it improved in the not-too-distant future, possibly version 11.1. (As a matter of general policy we do not make comments like that. I'm making an exception mostly as a way to hold myself to it.) – Daniel Lichtblau Aug 31 '16 at 15:04
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