Solutions of linear equations with singular parameter matrices

Question

I've been trying to solve a least square problem with a singular parameter matrix: solve x from $Ax=b$ with $A$ being singular. It has been confirmed that infinitely many solutions of $x$ in $Ax=b$ exist. Then suppose there is a disturbance $dA$ such that $(A+dA)$ is nonsingular, and $x'$ is solved from $x'=((A+dA)^T(A+dA))^{-1}(A+dA)^Tb$. Now with $dA$ approaching zero, what will happen to $x'$? Will it converge to one certain solution of $Ax=b$?