  We investigate the connection between the bubble-resummation and
critical-point methods for computing the $\beta$-functions in the limit of
large number of flavours, $N$, and show that these can provide complementary
information. While the methods are equivalent for single-coupling theories, for
multi-coupling case the standard critical exponents are only sensitive to a
combination of the independent pieces entering the $\beta$-functions, so that
additional input or direct computation are needed to decipher this missing
information. In particular, we evaluate the $\beta$-function for the quartic
coupling in the Gross-Neveu-Yukawa model, thereby completing the full system at
$\mathcal{O}(1/N)$. The corresponding critical exponents would imply a
shrinking radius of convergence when $\mathcal{O}(1/N^2)$ terms are included,
but our present result shows that the new singularity is actually present
already at $\mathcal{O}(1/N)$, when the full system of $\beta$-functions is
known.
