Let $f(x)$ be the polynomial with respect to $x,$ and $g_{n}(x)=n-2n^{2}\left|x-\frac{1}{2}\right|+\left|n-2n^{2}\left|x-\frac{1}{2}\right|\right|.$
Find $\lim_{n\to\infty}\int_{0}^{1}f(x)g_{n}(x)\ dx.$