The impact of different assumptions about high energy behavior of the total cross section of proton-proton interactions on the ratio of the real to imaginary part of the forward elastic scattering amplitude is analyzed. It is shown how experimental data about this ratio at LHC energies can help in the proper choice of the asymptotic dependence of the total cross section.The total cross sections and the ratios of real to imaginary parts of forward scattering amplitudes are tightly connected by dispersion relations. Hence, both of them should be analyzed simultaneously. We consider the interrelation of their behaviors as functions of energy discussing how different assumptions about the energy dependence of the total cross section σ t influence the predictions of the values of the ratiowhere A(s, t) is the elastic scattering amplitude at energy √ s and transferred momentum t. In its turn, this ratio measured at LHC can give some hints to the proper choice of the asymptotic behavior of the total cross section.Experimentally, this ratio is determined from the interference of the Coulomb and nuclear parts of the amplitude at extremely small angles. It is negative at comparatively low energies, becomes positive at energies √ s exceeding tens of GeV, and increases at ISR. These features are well reproduced by dispersion relations. The total cross sections of hadron processes rise with energy increase. This surprising fact was first noticed in kaon-proton interactions in Protvino [1]. Later, it was confirmed at ISR, SppS, and Tevatron. Nowadays, with advent of LHC, it has obtained further support.