I would like to calculate $\lim_ {n \to \infty} {\frac{n+\lfloor \sqrt{n} \rfloor^2}{n-\lfloor \sqrt{n} \rfloor}}$

Question

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.

Answer 1

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 $$ \left|\frac{(\lfloor \sqrt{n} \rfloor-\sqrt{n})(\lfloor \sqrt{n} \rfloor+\sqrt{n})}{n}\right|\leq\frac{2\sqrt{n}}{n} \to 0. $$

Answer 2

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.)