We investigate convergence of sequences of n x n matrix exponential functions t -> e(k)(tA-1) for t > 0, where A(k) -> A, A(k) is nonsingular and A is nilpotent. Specifically, we address pointwise convergence, almost uniform convergence, and, viewing the exponential as a Schwartz distribution, weak* convergence. We show that simple results can be obtained in terms of the eigenvalues of A(k)(-1) alone. In particular, a necessary and sufficient condition for weak* convergence in terms of eigenvalue behavior is attainable. We then apply our results to real-analytic matrices A(epsilon) as epsilon -> 0(+). Our work is applicable to matrices over both R and C.