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Very new to Mathematica so apologies for what is probably numpty question. The following returns no solutions. However, there is a solution! If I set a = 1 I get the solution b = 1/2. Can you explain how I'm misusing Solve?

f[x_] := Log[1/x + 1]^(-1);
Solve[Limit[Abs[f[x] - (x + b)], x -> Infinity] == 0, b]
(* out: {{b -> 1/2}} *)

contradicts

f[x_] := Log[1/x + 1]^(-1);
Solve[Limit[Abs[f[x] - (a x + b)], x -> Infinity] == 0, {a, b}]
(* out: {} *)
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Maybe because you have value for the limit only when a=1 check this:

    Limit[Abs[f[x] - (a x + b)], x -> Infinity, 
 Assumptions -> {a == 1, b \[Element] Reals}]

(*1/2 Abs[1 - 2 b]*)

Limit[Abs[f[x] - (a x + b)], x -> Infinity, 
 Assumptions -> {a != 1, {a, b} \[Element] Reals}]
(*\[Infinity]*)
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  • $\begingroup$ So why does solve not find this unique solution? It says there is no solution by returning {} $\endgroup$ Dec 24, 2014 at 22:22
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    $\begingroup$ @kungfujam Have a look At SolveAlways $\endgroup$ Dec 24, 2014 at 23:28

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