In the first part of the talk, we will discuss how asymptotic relations are preserved when integrated or differentiated in the sense of fractional operators. We will analyze fractional extension of the Karamata integration theorem, of the monotone density theorem, and of the generalized L'Hospital rule, among others. The second part of the talk will be devoted to applications of these tools in asymptotic theory of nonlinear fractional differential equations. This is joint work with Serena Matucci (University of Florence).