  The theory of limit operators was developed by Rabinovich, Roch and
Silbermann to study the Fredholmness of band-dominated operators on
$\ell^p(\mathbb{Z}^N)$ for $p \in \{0\} \cup [1,\infty]$, and recently
generalised to discrete metric spaces with property A by \v{S}pakula and
Willett for $p \in (1,\infty)$. In this paper, we study the remained extreme
cases of $p \in\{0,1,\infty\}$ (in the metric setting) to fill the gaps.
