= Solution
The hypothesis puts zero in the common unbounded resolvent component from part (iii). Hence $0\in\rho_B(x)$ for $B=\overline{\mathbb C[1,x]}$, and $(-x)^{-1}\in B$, or equivalently $x^{-1}\in B$. By the definition of this <norm> <closure>, for each $n$ choose a <polynomial> $p_n$ with $\|p_n(x)-x^{-1}\|<1/n$. Therefore
$$
\boxed{p_n(x)\longrightarrow x^{-1}.}
$$
This is an approximation in the algebra <norm>, not merely a scalar approximation on the <Banach algebra spectrum>.
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