  For each $n$, let $X_n \in \{0,\ldots,n\}$ be a random variable with mean
$\mu_n$, standard deviation $\sigma_n$, and let \[ P_n(z) = \sum_{k=0}^n
\mathbb{P}( X_n = k) z^k ,\] be its probability generating function. We show
that if none of the complex zeros of the polynomials $\{ P_n(z)\}$ are
contained in a neighbourhood of $1 \in \mathbb{C}$ and $\sigma_n >
n^{\varepsilon}$ for some $\varepsilon >0$, then $ X_n^* =(X_n -
\mu_n)\sigma^{-1}_n$ tends to a normal random variable $Z \sim
\mathcal{N}(0,1)$ in distribution as $n \rightarrow \infty$. Moreover, we show
this result is sharp in the sense that there exist sequences of random
variables $\{X_n\}$ with $\sigma_n > C\log n $ for which $P_n(z)$ has no roots
near $1$ and $X_n^*$ is not asymptotically normal. These results disprove a
conjecture of Pemantle and improve upon various results in the literature. We
go on to prove several other results connecting the location of the zeros of
$P_n(z)$ and the distribution of the random variables $X_n$.
