The purpose of this thesis is to develop a sufficient condition for a function holomorphic in a half-plane to have asymptotic values at a dense set of points on the boundary. The condition involves the convergence of an integral on a set of lines whose end points are dense on the bounding axis. An example is given to show that allowing various inclinations of the lines in the sufficiency condition is stronger than only allowing a fixed inclination. A second example shows that the condition given is not necessary.