It is proved in a rigorous way that the x → ∞ limits of the Mie normalized scattering cross section Qsc(x, m) and normalized extinction cross section Qcxt(x, m) are equal to 1+|(m - 1)/(m + 1)|2 and to 2, respectively.
Petr Chýlek, "Asymptotic limits of the Mie-scattering characteristics," J. Opt. Soc. Am. 65, 1316-1318 (1975)