Is it possible to study the behavior of the floor function at infinity by estimating its growth?
The floor function has countably many discontinuities at integers, so I'm afraid that these discontinuities will somehow make it hard to study its behavior.
We all have seen in a first calculus course that $$\lim_{n \to \infty} \lfloor n \rfloor = n + \text{ other negligible terms }$$
But is it possible to make this estimation more accurate as $n \to \infty$?
The reason that I'm asking this is because of this post of mine in here.
I'm trying to show that the following limit exists and is equal to $r$:
$$\lim_{n \to \infty} \sqrt{\lfloor (n+r)^2 \rfloor+1}-\sqrt{n^2}$$
I've already shown that:
$$\lim_{n \to \infty} \sqrt{\lfloor (n+r)^2 \rfloor+1}-\sqrt{n^2} = \lim_{n \to \infty}\frac{\lfloor (n+r)^2 \rfloor+1-n^2}{\sqrt{\lfloor (n+r)^2 \rfloor+1}+\sqrt{n^2}} = \lim_{n \to \infty} \frac{\lfloor (n+r)^2 \rfloor - n^2}{2n}$$
But now I'm stuck because now I need an asymptotic expansion of $\lfloor x \rfloor$ at infinity that gives better estimates.
Any ideas are appreciated.