  We suggest to use polarized nuclear targets of $^7$Li and $^3$He to study
nuclear effects in the spin dependent structure functions $g_{1A}(x,Q^2)$.
These effects are expected to be enhanced by a factor of two as compared to the
unpolarized targets.
  We predict a significant $x$ dependence at $10^{-4} \div 10^{-3} \leq x \leq
0.2$ of $g_{1A}(x,Q^2)/g_{1N}(x,Q^2)$ due to nuclear shadowing and nuclear
enhancement. The effect of nuclear shadowing at $x \approx 10^{-3}$ is of an
order of 16% for $g_{1A=7}^{n.s. 3/2}(x,Q^2)/g_{1N}^{n.s.}(x,Q^2)$ and 10% for
$g_{1A=3}^{n.s}(x,Q^2)/g_{1N}^{n.s.}(x,Q^2)$. By imposing the requirement that
the Bjorken sum rule is satisfied we model the effect of enhancement.
  We find the effect of enhancement at $x \approx 0.125 (0.15)$ to be of an
order of $20 (55)%$ for $g_{1A=7}^{n.s. 3/2}(x,Q^2)/g_{1N}^{n.s.}(x,Q^2)$ and
$14 (40)%$ for $g_{1A=3}^{n.s}(x,Q^2)/g_{1N}^{n.s.}(x,Q^2)$, if enhancement
occupies the region $0.05 \leq x \leq 0.2$ ($0.1 \leq x \leq 0.2$). We predict
a 2% effect in the difference of the scattering cross sections of deep
inelastic scattering of an unpolarized projectile off $^7$Li with $M_{J}$=3/2
and $M_{J}$=1/2. We also show explicitly that the many-nucleon description of
deep inelastic scattering off $^7$Li becomes invalid in the enhancement region
$0.05 < x \leq 0.2$.
