Sebastiao e Silva introduced in [7] the "order of growth" of distributions, which fitted very well into his axiomatic approach to the distribution theory. This notion enabled him to define a limit of distributions (both at finite and infinite points), the Landau "oh" symbols for them, and most important, the definite integral which led naturally to the convolution and Fourier transformation. In this paper the "equivalence at infinity" (analysed in [10]) is compared with the "order of growth" of distributions, and using both notions an asymptotic expansion of distributions is applied to the distributional Stieltjes transformation in the sense of [3].