Effective-medium theory and inelastic neutron scattering are used to study the magnetic excitations in the Ising ferromagnet LiHoF4 near its magnetic-field-induced quantum phase transition (QPT). As expected, the dominant mode softens at the QPT, but reaches a finite, rather than vanishing, limit due to the hyperfine interaction. The experimental phase diagram and excitation spectra are well described by an effective-medium theory to first order in the 1/z expansion. There are some differences between theory and experiments, which may be due to domain walls, their dynamics and shape effects, and to magnetoelastic couplings.