Laplace Transforms · Superchallenge problem

A leaky tank receives a unit dose at every positive integer time: $y' + y = \displaystyle\sum_{n=1}^{\infty}\delta(t - n)$, $y(0) = 0$. Using the Laplace transform, find the limit of the level just before the $N$-th dose, $\displaystyle\lim_{N\to\infty}\,\lim_{t\to N^-} y(t)$.