If $\displaystyle f : [-1, 1] \rightarrow \mathbb{R}$ and $\displaystyle f'(0) = \lim_{n\rightarrow \infty} nf\left(\frac{1}{n}\right)$ and $\displaystyle f(0) = 0$, find the value of $\displaystyle \lim_{n\rightarrow \infty} \left\{\frac{2}{\pi}(n + 1)\arccos \left(\frac{1}{n}\right) - n\right\}$.