According to the Poisson limit theorem, if $np\to\lambda$, then $\text{Bin}(n,p)\to\text{Poisson}(\lambda)$ (all when $n\to\infty$).
Does that mean, in particular, that if $np\to 0$, the limit distribution is the atomic distribution of 0? That is, mean 0 and 0 variance, no matter how slow does $np$ tend to 0? All it matter for is the rate of the convergence of the distribution sequence?