You're completely right! As stated, the problem is ill posed, i.e. it has no unique solution, so we didn't solve it.

Instead, we solved a similar problem by introducing a new parameter, \alpha. It was useful because we gained a mathematical description that works for very large n and s, and which matches our intuition about the problem.

It is important to recognize, as you point out, that that taking limits does not solve the problem. It just elucidates why we can't solve it as stated.