  The Jacobi polynomials $\hat{P}_n^{(\alpha,\beta)}(x)$ conform the canonical
family of hypergeometric orthogonal polynomials (HOPs) with the two-parameter
weight function $(1-x)^\alpha (1+x)^\beta, \alpha,\beta>-1,$ on the interval
$[-1,+1]$. The spreading of its associated probability density (i.e., the
Rakhmanov density) over the orthogonality support has been quantified, beyond
the dispersion measures (moments around the origin, variance), by the algebraic
$\mathfrak{L}_{q}$-norms (Shannon and R\'enyi entropies) and the monotonic
complexity-like measures of Cram\'er-Rao, Fisher-Shannon and LMC (L\'opez-Ruiz,
Mancini and Calbet) types. These quantities, however, have been often
determined in an analytically highbrow, non-handy way; specially when the
degree or the parameters $(\alpha,\beta)$ are large. In this work, we determine
in a simple, compact form the entropic and complexity-like properties of the
Jacobi polynomials in the two extremal situations: ($n\rightarrow \infty$;
fixed $\alpha,\beta$) and ($\alpha\rightarrow \infty$; fixed $n,\beta$). These
two asymptotics are relevant \textit{per se} and because they control the
physical entropy and complexity measures of the high energy (Rydberg) and high
dimensional (pseudoclassical) states of numerous supersymmetric
quantum-mechanical systems.
