 Solution

The complement of a compact planar set has a unique unbounded component: outside a large disc lies a connected exterior, and every unbounded component meets that exterior. The Neumann series for $|\lambda |>\Vert x\Vert$ converges in both algebras, so this exterior belongs to ${\rho }_B(x)$. Part i retains the whole component of ${\rho }_A(x)$ containing it. That retained component is unbounded and therefore is also the unique unbounded component of ${\rho }_B(x)$. The two unbounded components coincide.