limiter=Function[{input,variable,limit},
FullSimplify[TrigToExp[
Simplify[Normal[FullSimplify[
input SeriesData[variable,limit,List[1],0,1,1]
]]]]//.Plus[a_,b_]:>a/;AsymptoticEqual[a+b,a,variable->limit]]
];

In[130]:= f[x_] := 1/x^2 Exp[1/x] (Exp[1/x] + 1)^-2;

In[131]:= limiter[f[x], x, 0]

Out[131]= Sech[1/x]/(2 x^2)

In[132]:= limiter[f[x], x, Infinity]

Out[132]= 1/(4 x^2)

In[133]:= limiter[f[x], x, a]

Out[133]= 1/(2 a^2 (1 + Cosh[1/a]))