exchange integral at large distances
The author of the presentation is talking about large, but not yet very large/infinite distance here. Note the annotations:
$$
\frac{e^2}{4 \pi \epsilon R} = k/R
$$
when $R \rightarrow \infty$: slowly to zero



$$
S_{AB}
$$
when $R \rightarrow \infty$: to zero



$$
\left&lt; 1s_A\left| k/r_A \right| 1s_B\right&gt;
$$

when $R \rightarrow \infty$: rapidly to zero



So there is some regime somewhere, in which the change of the entire expression is dominated by the change in $S_{AB}$. The verification of the correctness of the different notations is apparently left to the reader ...