= Solution
Put $\Delta r=r_2-r_1$. Expanding the quotient $S/T$ consistently to first order gives
$$
v_{\rm eff}
=1+\frac{M}{r_2}
-\frac{M}{\Delta r}\log\frac{r_2}{r_1}.
$$
Thus
$$
\boxed{v_{\rm eff}
=1+\frac{M}{r_2}
-\frac{M}{r_2-r_1}\log\frac{r_2}{r_1}}.
$$
Because $\log(r_2/r_1)/(r_2-r_1)\to1/r_2$ as $r_1\to r_2$,
$$
\boxed{\lim_{r_1\to r_2}v_{\rm eff}=1}.
$$
This local limit is the expected speed of a massless particle measured in a freely falling <local inertial frame>; the finite-interval effective velocity combines nonlocal length and clock comparisons.
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