tails\] of the limits obtained along these germs.

Limits of formal branches {#formalbranches}
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In this subsection we recall the notion of formal branches and define the ‘limit’ of a formal branch. The limit of a curve ${{\mathscr C}}$ will be expressed in terms of the limits of its formal branches.

Choose affine coordinates $(y,z)=(1:y:z)$ so that $p=(0,0)$, and let $\Phi(y,z)=F(1:y:z)$ be the generator for the ideal of ${{\mathscr C}}$ in these coordinates. Decompose $\Phi(y,z)$ in ${{\mathbb{C}}}[[y,z]]$: $$\Phi(y,z)=\Phi_1(y,z)\cdot\cdots\cdot \Phi_r(y,z)$$ with $\Phi_i(y,z)$ irreducible power series. These define the [*irreducible branches*]{} of ${{\mathscr C}}$ at $p$. Each $\Phi_i$ has a unique tangent line at $p$; if this tangent line is [*not*]{} $y=0$, by the Weierstrass preparation theorem we may write (up to a unit in ${{\mathbb{C}}}[[y,z]]$) $\Phi_i$ as a monic polynomial in $z$ with coefficients in ${{\mathbb{C}}}[[y]]$, of degree equal to the multiplicity $m_i$ of the branch at $p$ (cf. for example [@MR1836037], §6.7). If $\Phi_i$ [*is*]{} tangent to $y=0$, we may likewise write it as a polynomial in $y$ with coefficients in ${{\mathbb{C}}}[[z]]$; [*mutatis mutandis,*]{} the discussion which follows applies to this case as well.

Concentrating on the first case, let $$\Phi_i(y,z)\in {{\mathbb{C}}}[[y]][z]$$ be a monic polynomial of degree $m_i$, defining an irreducible branch of ${{\mathscr C}}$ at $p$, not tangent to $y=0$. Then $\Phi_i$ splits (uniquely) as a product of linear factors over the ring ${{\mathbb{C}}}[[y^*]]$ of power series with [*rational nonnegative*]{} exponents: $$\Phi_i(y,z)=\prod_{j=1}^{m_i} \left(z-f_{ij}(y)\right)\quad,$$ with each $f_{ij}(y)$ of the form $$f(y)=\sum_{k\ge 0} \gamma_{\lambda_k} y^{\lambda_k}$$ with $\lambda_k\in {{\mathbb{Q}}}$, $1\le \lambda_0<\lambda_1<\dots$, and $\gamma_{\lambda_k}\ne 0$. We call each such $z=f(y)$ a [*formal branch*]{} of ${{\mathscr C}}$ at $p$. The branch is [*tangent to $z=0$*]{} if the dominating e