 Using asymptotic notation this result can be restated as

This notation (and the theorem) does "not" say anything about the limit of the "difference" of the two functions as increases without bound.Instead, the theorem states that approximates in the sense that the relative error of this approximation approaches 0 as increases without bound.The prime number theorem is equivalent to the statement that the th prime number satisfies

the asymptotic notation meaning, again, that the relative error of this approximation approaches 0 as increases without bound.