  For a given point P in the group of K-rational points E(K) of an elliptic
curve, we consider the sequence of values (F_1(P),F_2(P),F_3(P),...) of the
division polynomials of E at P. If K is a finite field, we prove that the
sequence is periodic. If K/Q_p is a local field, we prove (under certain
hypotheses) that there is a power q=p^e so that for all m > 0, the limit as k
goes to infinity of F_{mq^{k}}(P) exists in K and is algebraic over Q(E). We
apply this result to prove an analogous p-adic limit and algebraicity result
for elliptic divisibility sequences.
