$\lim_{n\to\infty} n^2 \int_{0}^{1} \frac{x\sin{x}}{1+(nx)^3} \, \mathrm{d}x$
$|f_n(x)| \leq 0$ is certainly false. It is true that $f_n (x) \to 0$ for each $x$.
If $x\leq \frac 1 n$ then $|\frac {n^{2}x \sin x} {1+n^{3}x^{3}}|\leq n^{2}x \sin x\leq 1$ since $\sin x \leq x$. If $x >\frac 1n$ then $|\frac {n^{2}x \sin x} {1+n^{3}x^{3}}|\leq |\frac {n^{2}x \sin x} {n^{3}x^{3}}|\leq \frac 1 {nx}\leq 1$. Hence, $|f_n(x) | \leq 1$ in both cases. Hence, the limit of the integral is $0$ (by DCT).