Should `DiscreteLimit` work when the argument type is specified?

Is it expected that DiscreteLimit fails to evaluate whenever the argument type is specified? E.g.,

Clear[f]
f[x_] := 0
DiscreteLimit[f[x], x -> Infinity]  (* evaluated *)

Clear[f]
f[x_Integer] := 0
DiscreteLimit[f[x], x -> Infinity]  (* not evaluated *)

Clear[f]
f[x : (_Integer | _Real | Infinity)] := 0
DiscreteLimit[f[x], x -> Infinity]  (* not evaluated *)

Edit: MichaelE2’s comment implies that the following should work, and it does.

Clear[f]
f[x : (_Integer | _Symbol)] := 0
DiscreteLimit[f[x], x -> Infinity]  (* evaluated *)