I would like to calculate the following limit: $$\lim_ {n \to \infty} {\frac{n+\lfloor \sqrt{n} \rfloor^2}{n-\lfloor \sqrt{n} \rfloor}}$$ where $\lfloor x \rfloor$ is floor of $x$ and $x ∈ R$.
Now I know the result is $2$, but I am having trouble getting to it. Any ideas would be greatly appreciated.