algebraic curves have the same [*asymptotic behavior*]{}, the Hausdorff distance between them is finite. As a consequence of the results obtained in this paper, in [@paper2], we present an algorithm for computing all the [*generalized asymptotes*]{} of a real plane algebraic curve $\cal C$ defined implicitly. The algorithm is based on the notion of perfect curve that, intuitively speaking, defines a curve of degree $d$ that cannot be approached by any curve of degree less than $d$.

The structure of the paper is as follows: In Section 2, we present the terminology that will be used throughout this paper as well as some previous results. In Section 3, the notion of [*infinity branch*]{} is introduced and some important properties are proved. In Section 4, we provide the notions of [*convergent branches*]{} and [*approaching curves*]{}. In addition, we develop some results that characterize whether two plane algebraic curves approach each other. The results presented in this section will be used in Section 5, where an algorithm to compare the asymptotic behavior of two algebraic plane curves is developed. In addition, we prove that if two plane curves have the same asymptotic behavior, the Hausdorff distance between them is finite.

Preliminaries and 