How do I evaluate
$$\lim_{n\to\infty}\left(\sqrt{n^2-n+1}-\left\lfloor\sqrt{n^2 – n+1}\right\rfloor\right),n\in\Bbb N$$
Attempt:
I thought of using Squeeze theorem but that could not help.
Secondly, we know that $x- \lfloor x\rfloor=\{x\}$ where $\{\}$ denotes the fractional part function. But I am not sure how to actually evaluate limits involving the fractional part function.
Best Answer
Since $n - 1 < \sqrt{n^2 - n + 1} < n$, then\begin{align*} &\mathrel{\phantom{=}}{} \sqrt{n^2 - n + 1} - [\sqrt{n^2 - n + 1}] = \sqrt{n^2 - n + 1} - (n - 1)\\ &= \frac{n}{\sqrt{n^2 - n + 1} + (n - 1)} → \frac{1}{2}. \quad (n → ∞) \end{align*}