We consider the asymptotic behavior of the polarization process for polar codes when the blocklength tends to infinity. In particular, we study the asymptotics of the cumulative distribution P(Z(n) <= z), where Z(n) = Z(W-n) is the Bhattacharyya process, and its dependence on the rate of transmission R. We show that for a BMS channel W, for R < I(W) we have lim(n ->infinity) P(Z(n) <= 2(-2n/2+root nQ-1(RI(W))/2+o(root n))/2) =R and for R < 1 - I(W) we have lim(n ->infinity) P(Z(n) >= 1 - 2(-2n/2+root nQ-1/(R1-I(W))/2+o(root n)) = R, where Q(x) is the probability that a standard normal random variable exceeds x. As a result, if we denote by P-e(SC)(n, R) the probability of error using polar codes of block-length N = 2(n) and rate R < I(W) under successive cancellation decoding, then log(-log(P-e(SC)(n, R))) scales as n/2 +root n(Q-1)(RI(W))/2 + o(root n). We also prove that the same result holds for the block error probability using the MAP decoder, i. e., for log(-log(P-e(MAP)(n, R))).