I would like to calculate $\lim_ {n \to \infty} {\frac{n+\lfloor \sqrt{n} \rfloor^2}{n-\lfloor \sqrt{n} \rfloor}}$ 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.
 A: You may observe that, as $n \to \infty$,
$$
\begin{align}
 {\frac{n+\lfloor \sqrt{n} \rfloor^2}{n-\lfloor \sqrt{n} \rfloor}}&={\frac{2n+(\lfloor \sqrt{n} \rfloor-\sqrt{n})(\lfloor \sqrt{n} \rfloor+\sqrt{n})}{n-\lfloor \sqrt{n} \rfloor}}\\\\
&={\frac{2+(\lfloor \sqrt{n} \rfloor-\sqrt{n})(\lfloor \sqrt{n} \rfloor+\sqrt{n})/n}{1-\lfloor \sqrt{n} \rfloor/n}}
\\\\& \to 2
\end{align}
$$ since, as $n \to \infty$,
$$
\left|\frac{\lfloor \sqrt{n} \rfloor}{n}\right|\leq\frac{\sqrt{n}}{n} \to 0
$$ and
$$
\left|\frac{(\lfloor \sqrt{n} \rfloor-\sqrt{n})(\lfloor \sqrt{n} \rfloor+\sqrt{n})}{n}\right|\leq\frac{2\sqrt{n}}{n} \to 0.
$$
A: Noting that $\lfloor\sqrt{n}\rfloor \operatorname*{\sim}_{n\to\infty} \sqrt{n} = o(n)$, the denominator is $n+o(n)$. But similarly, the same equivalent implies that $\lfloor\sqrt{n}\rfloor^2 \operatorname*{\sim}_{n\to\infty} \sqrt{n}^2 = n$, so the numerator becomes $2n+o(n)$. Putting it together, the expression is
$$
\frac{n+\lfloor\sqrt{n}\rfloor^2}{n-\lfloor\sqrt{n}\rfloor} = \frac{2n+o(n)}{n+o(n)}\xrightarrow[n\to\infty]{} 2
$$ 
(the equivalents and $o(\cdot)$ are the usual Landau notations.)
