There's no such thing as "infinitely small" deviation. The Manhattan Distance limit line is the hypotenuse, and its length is the length of the hypotenuse. The question is not about the limit line, which does not exhibit any weird behavior. The question is about the sequence of approximations to that line: why the length of the approximations doesn't converge to the length of the limit. And the answer is that it doesn't have to, because even though the approximations are very similar to the limit line in one respect (geometric closeness), they are all very different from it in another respect (directions and angles of travel). If we had a sequence of approximations whose direction of travel converged correctly, the length would converge correctly too.
