  Let $f \colon X \to X$ be a continuous map on a compact metric space $X$ and
let $\alpha_f$, $\omega_f$ and $ICT_f$ denote the set of $\alpha$-limit sets,
$\omega$-limit sets and nonempty closed internally chain transitive sets
respectively. We show that if the map $f$ has shadowing then every element of
$ICT_f$ can be approximated (to any prescribed accuracy) by both the
$\alpha$-limit set and the $\omega$-limit set of a full-trajectory.
Furthermore, if $f$ is additionally c-expansive then every element of $ICT_f$
is equal to both the $\alpha$-limit set and the $\omega$-limit set of a
full-trajectory. In particular this means that shadowing guarantees that
$\overline{\alpha_f}=\overline{\omega_f}=ICT(f)$ (where the closures are taken
with respect to the Hausdorff topology on the space of compact sets), whilst
the addition of c-expansivity entails $\alpha_f=\omega_f=ICT(f)$. We progress
by introducing novel variants of shadowing which we use to characterise both
maps for which $\overline{\alpha_f}=ICT(f)$ and maps for which
$\alpha_f=ICT(f)$.
