Non-tangential Limits and Bounded Point Derivations on \(R^2(X)\)

We study \(L^2\) $L^2$ approximations of rational functions on the complex plane, focusing on bounded point derivations. We show that if there is a bounded point derivation at x and \(\{x_n\}\) $\{x_n\}$ is a sequence of points that converges non-tangentially to x, then the sequence of derivatives \(\{f^{\prime }(x_n)\}\) $\{f^{\prime }(x_n)\}$ is uniformly bounded for a large class of functions, specifically those functions which can be approximated by rational functions with poles off X in the \(L^2\) $L^2$ norm. A counterexample is constructed that shows that the hypothesis of non-tangential convergence cannot be removed.