sketching its graph as well as in studying its topology (see e.g. [@Gao], [@lalo], [@Hong] and [@Zeng]).

Intuitively speaking, the infinity branch of a real plane algebraic curve reflects the status of a curve at the points with sufficiently large coordinates. An infinity branch is associated to a projective place centered at an infinity point, and it can be [*parametrized*]{} by means of Puiseux series. We show how to obtain this parametrization.

The concept of infinity branch allows us to introduce the notion of convergent branches and approaching curves. Intuitively speaking, two infinity branches converge if they get closer as they tend to infinity. This notion allows us to analyze whether two given implicit algebraic plane curves approach each other at the infinity.

More precisely, we say that a curve $\overline{\cal C}$ [*approaches*]{} ${\cal C}$ at its infinity branch $B$ if the distance between $\overline{\cal C}$ and $B$ approaches zero as they tend to infinity. We provide some results that characterize whether two plane algebraic curves are approaching.

Using these results, we present a method to compare the asymptotic behavior of two curves (i.e. the behavior of two curves at the infinity). In particular, we prove that if two plane 