The default range of Feints is ambiguous, but it was likely intended for melee.
By RAW, Feint can be done at any distance, but only provides melee benefits.
As you pointed out, the Core Rulebook does not explicitly say that Feints must be performed in melee range. The rules of Feinting don't involve a melee weapon or a combat maneuver check. However, the benefits of a successful feint only apply to melee attacks; the target is not denied their Dex bonus against ranged attacks by default.
Feinting is a standard action. To feint, make a Bluff skill check. The DC of this check is equal to 10 + your opponent's base attack bonus + your opponent's Wisdom modifier. If your opponent is trained in Sense Motive, the DC is instead equal to 10 + your opponent's Sense Motive bonus, if higher. If successful, the next melee attack you make against the target does not allow him to use his Dexterity bonus to AC (if any). This attack must be made on or before your next turn.
These rules were written around 2009 when Pathfinder 1e was first released, although the later revisions of the Core Rulebook never addressed this directly.
But, Feint was probably intended to be a melee action by default.
While the Core Rulebook text doesn't support this claim, and later errata doesn't specify it either, there is evidence that it is the case. The Paizo authors likely neglected to clarify this in their rulebook because their editing process is notoriously sloppy.
- The feat Ranged Feint from the Ultimate Intrigue expansion (2015) explicitly that feints are normally only usable with a melee weapon.
Normal: You can feint only with a melee weapon, and only against a creature you threaten with that weapon.
- The archer fighter archetype (Advanced Players Guide, 2010) gets a special ability to perform feints with a bow within 30 feet. While the wording is a little strange, it does imply that feinting from a distance is an exception rather than the normal rule.
At 3rd level, an archer can choose one of the following combat maneuvers or actions: disarm, feint, or sunder. He can perform this action with a bow against any target within 30 feet, with a –4 penalty to his CMB.
In a paizo.com discussion thread, the author Jason Nelson clarifies this further:
I wrote the APG fighter archetypes, and the implication I intended to convey was that you could both perform the feint and BENEFIT from the feint with your bow. It would seem a little silly to let them do the feint but not get any advantage from it, but if the RAW seems overly ambiguous and you're looking for RAI that's it.
- Pathfinder 1st edition was originally based on Dungeons & Dragons 3.5, and many of the rules in the Core Rulebook are directly copied from the original system without copying all the explanations. In the D&D Player's Handbook (p. 155), feinting is explicitly limited to melee range:
As a standard action, you can try to mislead an opponent in melee combat so that he can't dodge your next attack effectively. To feint, make a Bluff check opposed by a Sense Motive check by your target.
The full section on Feinting in D&D 3.5e is almost the same as in Pathfinder 1e.