Every serious source I consulted, be it Cajori, Struik, Edwards,... discusses the method of exhaustion as the means used by ancient Greeks to avoid “taking limits”, because they “disliked infinity”.
As far as my sources go, exhaustion is defined as proving that a sequence of values gets arbitrarily close to some value L by “doing a double reductio ad absurdum to guarantee the sequence does not get arbitrarily close to something less than L, nor something more than L”.
Then they all go on to say that this “avoids the step of taking a limit”. HOWEVER, the definition sounds exactly like what the modern definition of limit is! The only practical difference I see is the consistent use of reductio to reach the result.
What subtlety am I missing?