Question: finding constans of a function

hi

i have the following function

 

f[1](x):=_C2 (e)^x-_C1 (e)^(-x)+1/2 x (e)^(-x) h1+1/2 (e)^(-x) h1+_C3

with thease boundary conditions:   f[1](x))=0,diff(f[1](x),x))=1,diff(f[1](x),x),x=infinity)

i want to find cinstants C1,C2,C3 where h1 is arbitrary parametr.

i write the following code but its doesnt work correctly

s1:=simplify(subs(x=0,f[1](x))=0);
s2:=simplify(subs(x=0,diff(f[1](x),x))=1);
s3:=simplify(limit(diff(f[1](x),x),x=infinity))=0;
s:={s1,s2,s3};
solve(s,{_C1,_C2,_C3});

the results are:

s1 := _C2-_C1+(1/2)*h1+_C3 = 0
s2 := _C2+_C1 = 1
s3 := signum(_C2)*infinity = 0
> s := {signum(_C2)*infinity = 0, _C2+_C1 = 1, _C2-_C1+(1/2)*h1+_C3 = 0};

  as you can see C2 must be zero bud it doesnt. where is my problem. please help

thancks

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