Who can give me some function with asymptotics ?

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SUMMARY

The discussion centers on finding a simpler function f(r) with specific asymptotic behavior as x approaches 0 and infinity. The user proposes a complex function A/B*Log[1+x], where A=1+x/2+x^3/12 and B=1+d*x^2, but seeks a more straightforward alternative. The desired asymptotic forms are f -> 1*x + a*x^3 + b*x^5 + c*x^7 as x -> 0 and f -> x*Log[x] as x -> infinity. The function 1/x is mentioned as a valid option, with asymptotes at the y and x axes.

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sufive
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Dear Everybody,

I need some some function f(r) with asymptotics

as x -> 0, f-> 1*x+a*x^3+b*x^5+c*x^7+..., a,b,c>0
as x -> infinity, f-> x*Log[x]

I construct one like A/B*Log[1+x], where A=1+x/2+x^3/12, B=1+d*x^2.
But I think it is too complex, such that I cannot continue my successive latter works.

Can somebody give me a more simple such functions?

Suggestion or hints are also welcome.

Thank you very much!
 
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1/x works. Asymptotes are the y and x axes.
 

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