Using polydispersity index as an additional order parameter we investigate freezing/melting transition of Lennard-Jones polydisperse systems (with Gaussian polydispersity in size), especially to gain insight into the origin of the terminal polydispersity. The average inherent structure (IS) energy and root mean square displacement (RMSD) of the solid before melting both exhibit quite similar polydispersity dependence including a discontinuity at solid-liquid transition point. Lindemann ratio, obtained from RMSD, is found to be dependent on temperature. At a given number density, there exists a value of polydispersity index (𝛿_P) above which no crystalline solid is stable. This transition value of polydispersity (termed as transition polydispersity, 𝛿_P) is found to depend strongly on temperature, a feature missed in hard sphere model systems. Additionally, for a particular temperature when number density is increased, 𝛿_P shifts to higher values. This temperature and number density dependent value of 𝛿_P saturates surprisingly to a value which is found to be nearly the same for all temperatures, known as terminal polydispersity (𝛿_TP). This value (𝛿_TP ∼ 0.11) is in excellent agreement with the experimental value of 0.12, but differs from hard sphere transition where this limiting value is only 0.048. Terminal polydispersity (𝛿_TP) thus has a quasiuniversal character. Interestingly, the bifurcation diagram obtained from non-linear integral equation theories of freezing seems to provide an explanation of the existence of unique terminal polydispersity in polydisperse systems. Global bond orientational order parameter is calculated to obtain further insights into mechanism for melting.