= Solution
For a slowly moving particle in the <weak-field approximation>, $dt/d\tau=1+O(\Phi,v^2)$ and terms containing two spatial velocities may be neglected. The spatial <geodesic equation> reduces to
$$
\frac{d^2x^i}{dt^2}=-\Gamma^i{}_{00}.
$$
For $g_{00}=-(1+2\Phi)$, the <Levi-Civita connection> is
$$
\Gamma^i{}_{00}
=\frac12g^{ij}(-\partial_jg_{00})
=\partial_i\Phi+O(\Phi^2).
$$
Hence the <Newtonian limit of general relativity> is
$$
\boxed{\frac{d^2\mathbf x}{dt^2}=-\nabla\Phi}.
$$
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