  We show that, under a natural scaling, the small-time behavior of the
logarithmic derivatives of the subelliptic heat kernel on $SU(2)$ converges to
their analogues on the Heisenberg group at time 1. Realizing $SU(2)$ as
$\mathbb{S}^3$, we then generalize these results to higher-order
odd-dimensional spheres equipped with their natural subRiemannian structure,
where the limiting spaces are now the higher-dimensional Heisenberg groups.
