There are several examples in the literature of polynomial families that converge uniformly on compact sets to families of transcendental functions. In joint work with Jane Hawkins and Lorelei Koss, we have a family of rational functions that converges to a transcendental family. This transcendental family contains parameters for which the Fatou set is empty. Since a rational function can have an empty Fatou set but a polynomial cannot, this approximation by rational functions is, in a sense, more natural than the previously known approximations by polynomials. Speaker(s): Joanna Furno (IUPUI)