  The axial charge of the nucleon $g_A$ and the pion decay constant $f_\pi$ are
computed in two-flavor lattice QCD. The simulations are carried out on lattices
of various volumes and lattice spacings. Results are reported for pion masses
as low as $m_\pi=130\,\mbox{MeV}$. Both quantities, $g_A$ and $f_\pi$, suffer
from large finite size effects, which to leading order ChEFT and ChPT turn out
to be identical. By considering the naturally renormalized ratio $g_A/f_\pi$,
we observe a universal behavior as a function of decreasing quark mass. From
extrapolating the ratio to the physical point, we find $g_A^R=1.29(5)(3)$,
using the physical value of $f_\pi$ as input and $r_0=0.50(1)$ to set the
scale. In a subsequent calculation we attempt to extrapolate $g_A$ and $f_\pi$
separately to the infinite volume. Both volume and quark mass dependencies of
$g_A$ and $f_\pi$ are found to be well decribed by ChEFT and ChPT. We find at
the physical point $g_A^R=1.24(4)$ and $f_\pi^R=89.6(1.1)(1.8)\,\mbox{MeV}$.
Both sets of results are in good agreement with experiment. As a by-product we
obtain the low-energy constant $\bar{l}_4=4.2(1)$.
