A 55 millimeter binocular aperture compared to a fully dilated 4 millimeter human eye has an area that is 189 times larger. This means that its light collecting ability is 189 times greater and the magnitude limit can be calculate from 189 = 10^(0.4m). The answer is that if the eye can just see to a magnitude of + 6.3, then the binocular can see +5.7 magnitudes fainter to +12.0.
A 4.5 inch telescope ( 114.3 millimeters diameter) has 816.5 times the light gathering area of the human eye, so the magnitude limit is +6.3 + 7.3 = +13.6m.