Consider the stationary measure of open asymmetric simple exclusion process (ASEP) on the lattice $\{1,\dots,n\}$. Taking $n$ to infinity while fixing the jump rates, this measure converges to a measure on the semi-infinite lattice. In the high and low density phases, we characterize the limiting measure and also show that this convergence occurs in total variation distance on a sublattice of scale $n/\log n$. Our approach involves bounding the total variation distance using generating functions, which are further estimated through a subtle analysis of the atom masses of Askey--Wilson signed measures. (1).png)