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  1. limiter=Function[{input,variable,limit},
  2. FullSimplify[TrigToExp[
  3. Simplify[Normal[FullSimplify[
  4. input SeriesData[variable,limit,List[1],0,1,1]
  5. ]]]]//.Plus[a_,b_]:>a/;AsymptoticEqual[a+b,a,variable->limit]]
  6. ];
  7.    
  8. In[130]:= f[x_] := 1/x^2 Exp[1/x] (Exp[1/x] + 1)^-2;
  9.  
  10. In[131]:= limiter[f[x], x, 0]
  11.  
  12. Out[131]= Sech[1/x]/(2 x^2)
  13.  
  14. In[132]:= limiter[f[x], x, Infinity]
  15.  
  16. Out[132]= 1/(4 x^2)
  17.  
  18. In[133]:= limiter[f[x], x, a]
  19.  
  20. Out[133]= 1/(2 a^2 (1 + Cosh[1/a]))
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