  This article is about finding the limit set for banded Toeplitz matrices. Our
main result is a new approach to approximate the limit set $\Lambda(b)$ where
$b$ is the symbol of the banded Toeplitz matrix. The new approach is
geometrical and based on the formula $\Lambda(b) = \cap_{\rho \in (0, \infty)}
\text{sp } T(b_\rho)$. We show that the full intersection can be approximated
by the intersection for a finite number of $\rho$'s, and that the intersection
of polygon approximations for $\text{sp } T(b_\rho)$ yields an approximating
polygon for $\Lambda(b)$ that converges to $\Lambda(b)$ in the Hausdorff
metric. Further, we show that one can slightly expand the polygon
approximations for $\text{sp } T(b_\rho)$ to ensure that they contain $\text{sp
} T(b_\rho)$. Then, taking the intersection yields an approximating superset of
$\Lambda(b)$ which converges to $\Lambda(b)$ in the Hausdorff metric, and is
guaranteed to contain $\Lambda(b)$. We implement the algorithm in Python and
test it. It performs on par to and better in some cases than existing
algorithms. We argue, but do not prove, that the average time complexity of the
algorithm is $O(n^2 + mn\log m)$, where $n$ is the number of $\rho$'s and $m$
is the number of vertices for the polygons approximating $\text{sp }
T(b_\rho)$. Further, we argue that the distance from $\Lambda(b)$ to both the
approximating polygon and the approximating superset decreases as
$O(1/\sqrt{k})$ for most of $\Lambda(b)$, where $k$ is the number of elementary
operations required by the algorithm.
