  We present calculations of certain limits of scheme-independent series
expansions for the anomalous dimensions of gauge-invariant fermion bilinear
operators and for the derivative of the beta function at an infrared fixed
point in SU($N_c$) gauge theories with fermions transforming according to two
different representations. We first study a theory with $N_f$ fermions in the
fundamental representation and $N_{f'}$ fermions in the adjoint or symmetric or
antisymmetric rank-2 tensor representation, in the limit $N_c \to \infty$, $N_f
\to \infty$ with $N_f/N_c$ fixed and finite. We then study the $N_c \to \infty$
limit of a theory with fermions in the adjoint and rank-2 symmetric or
antisymmetric tensor representations.
