have that $\lim_{l\rightarrow\infty}b_{n_l}/a_{n_l}=\infty$, and then $\lim_{l\rightarrow\infty}a_{n_l}/b_{n_l}=0$. Hence, $$\lim_{l\rightarrow\infty}F(Q_{n_l})=F(0:1:0)=0,\qquad Q_{n_l}=(a_{n_l}/b_{n_l}:1:1/b_{n_l})$$ which implies that $P=(0:1:0)$ is an infinity point of ${\cal C}^*$.

2.  If there exists a monotone subsequence $\{b_{n_l}/a_{n_l}\}_{l\in {\Bbb N}}$ that is bounded, we have that $\lim_{l\rightarrow\infty}b_{n_l}/a_{n_l}=m$. Thus, $$\lim_{l\rightarrow\infty}F(Q_{n_l})=F(1:m:0)=0,\qquad Q_{n_l}=(1:b_{n_l}/a_{n_l}:1/a_{n_l})$$ which implies that $P=(1:m:0)$ is an infinity point of ${\cal C}^*$.

From both situations, we conclude that there exist a sequence $\{Q_n\}_{n\in {\Bbb N}}$ that approaches to an infinity point $P$ as $n$ tends to infinity; that is, there exists $M\in {\Bbb R}^+$ such that $\|Q_n-P\|\leq \epsilon$, for $n\geq M$. Thus, we deduce that $\{Q_n\}_{n\in {\Bbb
N},\,n\geq M}$ can be obtained by a place centered at $P$. Hence, $p_n$ belongs to some infinity branch of ${\cal C}$, which contradicts the hypothesis. $\Box$

\[R-BolaMaxima\] Reasoning similarly as in Lemma \[L-BolaMaxima\], one has that there exists $K\in\mathbb{R}^+$ such that for every $p=(a,b)\in\mathcal{C}$ with $|b|>K$, it holds that $p\in B_p$, where $B_p$ is an infinity branch of ${\cal C}$.

Convergent Branches and\
Approaching Curves
========================

In this section, we introduce the notions of convergent branches and approaching curves. Intuitively speaking, two infinity branches converge if they get closer as they tend to infinity. This concept will allow us to analyze whether two curves 