Let \(f\) be a holomorphic function in the unit disc omitting a set of values \(A\) of the complex plane. If \(A\) has positive logarithmic capacity, R. Nevanlinna proved that \(f\) has radial limit at almost every point of the unit circle, while if a closed set \(A\) has zero logarithmic capacity, then there is \(f\) omitting \(A\) with radial limits almost nowhere in the unit circle. Nonetheless, we prove that if \(A\) is an infinite set, then \(f\) has limit along a set of radii which has Hausdorff dimension 1. A localization technique reduces this result to the following theorem concerning inner functions. Let \(I\) be an inner function omitting a set of values \(B\) in the unit disc, then for any accumulation point \(b\) of \(B\) in the unit disc, there exists a set of Hausdorff dimension 1 of radii along which the function \(I\) has limit \(b\).