Let  … …be the Engel expansion of an irrational number x. One of the major subjects in the study of Engel expansions is to consider the growth rate of for almost all x in a certain sense. It was announced by Borel (1947) that for Lebesgue almost all x, the limit of equals to 1. In this talk, we are concerned with the limit behaviors of log – and in the senses of Baire classification and Hausdorff dimension, where is a positive function.