 Bennett bounds for sparse binomial fluctuations

For a binomial distribution with $p_n\to 0$ and $n{p}_n\to \infty$, the Bennett inequality at $C$ standard deviations tends to $2e^{-C^2/2}$, whereas the Hoeffding inequality tends to two. The raw Bennett inequality bound is eventually smaller for any fixed $C>0$; after clipping at one it is nontrivial in the limit only when $C>\sqrt{2\log 2}$.