  Motivated by recent attention to a potential antiferromagnetic quantum
critical point at $x_c\sim 0.19$, we have used inelastic neutron scattering to
investigate the low-energy spin excitations in crystals of
La$_{2-x}$Sr$_x$CuO$_4$ bracketing $x_c$. We observe a peak in the normal-state
spin-fluctuation weight at $\sim20$~meV for both $x=0.21$ and 0.17,
inconsistent with quantum critical behavior. The presence of the peak raises
the question of whether low-energy spin fluctuations limit the onset of
superconducting order. Empirically evaluating the spin gap $\Delta_{\rm spin}$
in the superconducting state, we find that $\Delta_{\rm spin}$ is equal to the
coherent superconducting gap $\Delta_c$ determined by electronic
spectroscopies. To test whether this is a general result for other cuprate
families, we have checked through the literature and find that
$\Delta_c\le\Delta_{\rm spin}$ for cuprates with uniform $d$-wave
superconductivity. We discuss the implications of this result.
