  An iterative optimization method applied to a function $f$ on $\mathbb{R}^n$
will produce a sequence of arguments $\{\mathbf{x}_k\}_{k \in \mathbb{N}}$;
this sequence is often constrained such that $\{f(\mathbf{x}_k)\}_{k \in
\mathbb{N}}$ is monotonic. As part of the analysis of an iterative method, one
may ask under what conditions the sequence $\{\mathbf{x}_k\}_{k \in
\mathbb{N}}$ converges. In 2005, Absil et al.\ employed the {\L}ojasiewicz
gradient inequality in a proof of convergence; this requires that the objective
function exist at a cluster point of the sequence. Here we provide a
convergence result that does not require $f$ to be defined at the limit
$\lim_{k \to \infty} \mathbf{x}_k$, should the limit exist. We show that a
variant of the {\L}ojasiewicz gradient inequality holds on sets adjacent to
singularities of bounded multivariate rational functions. We extend the results
of Absil et al.\ to prove that if $\{\mathbf{x}_k\}_{k \in \mathbb{N}} \subset
\mathbb{R}^n$ has a cluster point $\mathbf{x}_*$, if $f$ is a bounded
multivariate rational function on $\mathbb{R}^n$, and if a technical condition
holds, then $\mathbf{x}_k \to \mathbf{x}_*$ even if $\mathbf{x}_*$ is not in
the domain of $f$. We demonstrate how this may be employed to analyze divergent
sequences by mapping them to projective space, and consider the implications
this has for the study of low-rank tensor approximations.
