Let $V$ be an analytic variety in some open set in ${{\mathbb{C}}^{n}}$. For a real analytic curve $\gamma $ with $\gamma (0)\,=\,0$ and $d\,\ge \,1$, define ${{V}_{t}}\,=\,{{t}^{-d}}(V\,-\,\gamma (t))$. It was shown in a previous paper that the currents of integration over ${{V}_{t}}$ converge to a limit current whose support ${{T}_{\gamma ,d}}V$ is an algebraic variety as $t$ tends to zero. Here, it is shown that the canonical defining function of the limit current is the suitably normalized limit of the canonical defining functions of the ${{V}_{t}}$. As a corollary, it is shown that ${{T}_{\gamma ,d}}V$ is either inhomogeneous or coincides with ${{T}_{\gamma ,\,\delta }}V$ for all $\delta $ in some neighborhood of $d$. As another application it is shown that for surfaces only a finite number of curves lead to limit varieties that are interesting for the investigation of Phragmén-Lindelöf conditions. Corresponding results for limit varieties ${{T}_{\sigma ,\delta }}W$ of algebraic varieties $W$ along real analytic curves tending to infinity are derived by a reduction to the local case.