In this letter, a theoretical method for the analysis of diffusive flux/current to limited scale self-affine random fractals is presented and compared with experimentally measured electrochemical current for such roughness. The theory explains the several experimental findings of the temporal scale invariance as well as deviation from this of current transients in terms of three dominant fractal parameters for the limited-length scales of roughness. This theoretical method is based on limited scale power-law characterization of the interfacial roughness spectrum and the solution of diffusion equation under the diffusion-limited boundary conditions on rough interfaces. More broadly, these calculations challenges the earlier belief that the anomalous behavior is solely dependent on fractal dimension of roughness and highlight the potential to apply this equation for the scale invariant roughness determination. Finally, the validity of theoretical result is tested with extensive experimental data.