Some supernova remnant X-ray spectra show evidence for synchrotron emission from the extension of the electron spectrum that produces radio synchrotron emission. For any remnant, if the extrapolated radio flux exceeds the observed X-ray flux, thermal or nonthermal, a roll-off of the relativistic electron energy distribution must occur below X-ray-emitting energies. We have studied the X-ray emission from 11 remnants in the Large Magellanic Cloud to constrain this roll-off energy. We assume that the electron distribution is a power law with an exponential cutoff at some Emax and radiates in a uniform magnetic field. If the radio flux and spectral index are known, this simple model for the synchrotron contribution depends on only one parameter that relates directly to Emax. Here we have modeled the X-ray spectra by adding a component for thermal radiation of a Sedov blast wave to the synchrotron model. For all 11 supernova remnants in this sample, the limits for Emax range between 10 and 80 TeV (for a mean magnetic field of 10 μG). This result is similar to a study of Galactic remnants in which 13 out of 14 objects had limits between 20 and 80 TeV. We interpret Emax in the context of shock acceleration theories. Better data and models should allow either firm detections of nonthermal components or more restrictive limits on Emax.