for each $z_n$ such that $|z_n|>M$, $(p_n,q_n)$ is the point of ${\overline{{\cal C}}}$ closest to the point $(z_n,r(z_n))$ (this point exists because of Definition \[D-distance\]). Note that the above equality implies that $$\lim_{n\rightarrow\infty}|p_n-z_n|^2+|q_n-r(z_n)|^2=0,$$ and hence we have that:

-   $\displaystyle\lim_{n\rightarrow\infty}(p_n-z_n)=0$. Then, $\lim_{n\rightarrow\infty} p_n/z_n=1$ which implies that $\lim_{n\rightarrow\infty} p_n=\infty$. Hence, $\lim_{n\rightarrow\infty}1/p_n=0.$

-   $\displaystyle\lim_{n\rightarrow\infty}(q_n-r(z_n))=0$. Then, $\lim_{n\rightarrow\infty}(q_n/z_n-r(z_n)/z_n)=0$ which implies that $\lim_{n\rightarrow\infty} q_n/z_n =\lim_{n\rightarrow\infty} r(z_n)/z_n =m.$

Therefore, $$\lim_{n\rightarrow\infty} q_n/p_n =\lim_{n\rightarrow\infty} \frac{q_n/z_n}{p_n/z_n} =m.$$ Now, taking into account Lemma \[L-BolaMaxima\] and that $\lim_{n\rightarrow\infty} p_n=\infty$, we get that there exits $n_0\in {\Bbb N}$ such that for $n\geq n_0$, the points $(p_n,q_n)$ are in some infinity branch of ${\overline{{\cal C}}}$. Moreover, since any curve has a finite number of infinity branches and a finite number of leaves, we can find a subsequence $\{z_{n_l}\}_{l\in {\Bbb N}}$ and $l_0\in {\Bbb N}$ such that for $l\geq l_0$, the points $(p_{n_l},q_{n_l})$ are all in a same leaf $\overline{L}=\{(z,\overline{r}(z))\in {\Bbb C}^2:\,z\in {\Bbb
C},\,|z|>\overline{M}\}$, belonging to some branch $\overline{B}\subset{\overline{{\cal C}}}$.

Under these conditions, we deduce that for $l\geq l_0$, $q_{n_l}=\overline{r}(p_{n_l})$, and then $$\displaystyle\lim_{l\rightarrow\infty}\overline{r}(p_{n_l})/p_{n_l}=\lim_{l\rightarrow\infty}q_{n_l}/p_{n_l}=m.$$ Since the limit $\lim_{z\rightarrow\infty}\frac{\overline{r}(z)}{z}=\lim_{z\rightarrow\infty}\overline{\varphi}(z^{-1})$ exists, we get $\lim_{z\rightarrow\infty}\frac{\overline{r}(z)}{z}=m$.

In addition, note that $$|r(z_{n_l})-\overline{r}(z_{n_l})|=d((z_{n_l},r(z_{n_l})),(z_{n_l},\overline{r}(z_{n_l})))\leq$$ $$d((z_{n_l},r(z_{n_l})),(p_{n_l},\overline{r}(p_{n_l})))
+d((p_{n_l},\overline{r}(p_{n_l})),(z_{n_l},\overline{r}(z_{n_l})))\qquad \mbox{(I)}$$ and $$d((z_{n_l},r(z_{n_l})),(p_{n_l},\overline{r}(p_{n_l})))=d((z_{n_l},r(z_{n_l})),(p_{n_l},q_{n_l}))_{\overrightarrow{l\rightarrow
\infty}}0.$$ Now, let us prove that $d((p_{n_l},\overline{r}(p_{n_l})),(z_{n_l},\overline{r}(z_{n_l})))_{\overrightarrow{l\rightarrow
\infty}}0.$ For this purpose, we show 