  While independent observations have been made regarding the behaviour of
effective quasinormal mode (QNM) potentials within the large angular momentum
limit, we demonstrate analytically here that a uniform expression emerges for
non-rotating, higher-dimensional, and spherically-symmetric black holes (BHs)
in this regime for fields of integer and half-integer spin in asymptotically
flat and dS BH contexts; a second uniform expression arises for these QNM
potentials in AdS BH spacetimes. We then proceed with a numerical analysis
based on the multipolar expansion method recently proposed by Dolan and
Ottewill to determine the behaviour of quasinormal frequencies (QNF) for
varying BH parameters in the eikonal limit. We perform a complete study of
Dolan and Ottewill's method for perturbations of spin $s \in \{0,1/2,1,3/2,2
\}$ in 4D Schwarzschild, Reissner-Nordstr{\"o}m, and Schwarzschild de Sitter
spacetimes, clarifying expressions and presenting expansions and results to
higher orders $(\mathcal{O}(L^{-6}))$ than many of those presented in the
literature $(\sim \mathcal{O}(L^{-2}))$. We find good agreement with known
results of QNFs for low-lying modes; in the large-$\ell$ regime, our results
are highly consistent with those of Konoplya's 6th-order WKB method. We confirm
a universality in the trends of physical features recorded in the literature
for the low-lying QNFs (that the real part grows indefinitely, the imaginary
tends to a constant as $\ell \rightarrow \infty$, etc.) as we approach large
values of $\ell$ within these spacetimes, and explore the consequent interplay
between BH parameters and QNFs in the eikonal limit.
