In order to illustrate this remark, we consider the curves ${\cal C}$ and $\overline{{\cal C}}$, defined by the polynomials $f(x,y)=y^4-2xy^2+x^2-y$, and $\overline{f}(x,y)=y^2-x$, respectively. ${\cal C}$ has only the infinity branch $B=\bigcup_{i=1}^4 L_i$, where $L_i=\{(z,r_i(z))\in {\Bbb C}^2:
\,z\in {\Bbb C},\,|z|>M\}$, $$r_i(z)=c_i^{2}
z^{1/2}+\frac{1}{2}c_i^{5}z^{-1/4}-\frac{1}{64}c_i^{11}z^{-7/4}+\frac{1}{128}c_i^{14}z^{-10/4}+\cdots,$$ and $c_1=1$, $c_2=I$, $c_3=-1$, $c_4=-I$. Note that the first term of these series is $z^{1/2}$ or $-z^{1/2}$. The curve $\overline{{\cal C}}$ also has one infinity branch defined by $\overline{B}=\bigcup_{i=1}^2 \overline{L}_i$, where $\overline{L}_i=\{(z,\overline{r}_i(z))\in {\Bbb C}^2: \,z\in {\Bbb
C},\,|z|>\overline{M}\}$, $\overline{r}_i(z)=d_i z^{1/2},$ and $d_1=1$, $d_2=-1$. We get that $B$ and $\overline{B}$, are convergent since $L_1$ and $\overline{L}_1$ converge. In fact, $L_1$ and $L_3$ converge with $\overline{L}_1$ and, on the other hand, $L_2$ and $L_4$ converge with $\overline{L}_2$ (see Lemma \[L-DistVertical\]).

Two convergent infinity branches may be contained in the same curve or they may belong to different curves. In this second case we will say that those curves [*approach each other*]{}. In order to define this concept in a more formal way, we first introduce the following distance:

\[D-distance\] Given an algebraic plane curve ${\cal C}$ over $\Bbb C$ and a point $p\in {\Bbb
C}^2$, we define [*the distance from $p$ to ${\cal C}$*]{} as $ d(p,{\cal C})=\min\{d(p,q):q\in{\cal C}\}.$

Observe that this 