Polynomial approximation obstruction on a punctured circle

Polynomials cannot converge uniformly to $1/z$ on $\{z:|z|=1,z\ne 1\}$. Uniform convergence there makes the polynomials uniformly Cauchy on the full circle because deleting one point does not change the supremum of a continuous function. They would therefore converge uniformly to $1/z$ on the full circle, contradicting