  We study the possibility of an ultraviolet (UV) zero in the $n$-loop beta
function of U(1) and non-Abelian gauge theories with $N_f$ fermions for large
$N_f$. The effect of scheme transformations on the coefficients of different
powers of $N_f$ in the $n$-loop term in the beta function is calculated. A
general scheme-independent criterion is given for determining whether or not
the $n$-loop beta function has a UV zero for large $N_f$. We compare the
results with exact integral representations of the leading terms in the beta
functions for the respective Abelian and non-Abelian theories in the limit $N_f
\to \infty$ limit with $N_f \alpha$ finite. As part of this study, new analytic
and numerical results are presented for certain coefficients, denoted
$b_{n,n-1}$, that control the large-$N_f$ behavior at $n$-loop order in the
beta function. We also investigate various test functions incorporating a
power-law and essential UV zero in the beta function and determine their
manifestations in series expansions in powers of coupling and in powers of
$1/N_f$.
