Leonin Arbiter says:
Players can't search libraries. Any player may pay {2} for that player to ignore this effect until end of turn.
If I cast Path to Exile on the Arbiter, its controller would usually be able to search for a basic land. Arbiter makes it so that they can't search the library unless they pay {2}, but the Path also removes Arbiter, and once it's no longer on the battlefield presumably its effect doesn't apply anymore. Can the Arbiter's controller search for a basic land or not?