  It has been argued from the earliest days of quantum chromodynamics (QCD)
that at asymptotically small values of $x$ the parton distribution functions
(PDFs) of the proton behave as $x^\alpha$, where the values of $\alpha$ can be
deduced from Regge theory, while at asymptotically large values of $x$ the PDFs
behave as $(1-x)^\beta$, where the values of $\beta$ can be deduced from the
Brodsky-Farrar quark counting rules. We critically examine these claims by
extracting the exponents $\alpha$ and $\beta$ from various global fits of
parton distributions, analysing their scale dependence, and comparing their
values to the naive expectations. We find that for valence distributions both
Regge theory and counting rules are confirmed, at least within uncertainties,
while for sea quarks and gluons the results are less conclusive. We also
compare results from various PDF fits for the structure function ratio
$F_2^n/F_2^p$ at large $x$, and caution against unrealistic uncertainty
estimates due to overconstrained parametrisations.
