Dynamic crack propagation in brittle solids is numerically analyzed using a two-dimensional finite element method based on the phase-field model of fracture to investigate the limiting crack velocity. Straight crack propagation without branching or kinking is achieved by modeling an elastically homogeneous and isotropic medium that contains a narrow rectangular region with a relatively low critical energy release rate. Mode I and mode II crack propagation are studied for two types of strain energy degradation functions in the phase-field model: quadratic and cubic forms. For both degradation functions, the mode I crack accelerates toward the Rayleigh wave velocity, which is the theoretically predicted limiting velocity according to linear elastic fracture mechanics. In contrast, the two degradation functions are found to yield different propagation characteristics for the mode II crack. With the quadratic degradation function, the crack velocity increases smoothly beyond the shear wave velocity. In the case of the cubic degradation function, however, the crack accelerates discontinuously from below the Rayleigh wave velocity to above the shear wave velocity due to the nucleation of a secondary crack ahead of the main crack tip, followed by coalescence of the secondary and main cracks. This behavior is consistent with theoretical predictions based on linear elastic fracture mechanics as well as previous molecular dynamics and finite difference simulations.