Let $y=-x$
$$\lim\_{y\to \infty} \frac{y^4 \sin (-\frac 1y) + y^2}{1+|y|^3}$$
$$=\lim\_{y\to \infty} \frac{y^3 ((-y \sin \frac 1y) +\frac 1y)}{y^3(1+\frac{1}{y^3})}$$

At $y\to \infty$, $y\sin \frac 1y=0$ (What I think is happening)

Therefore the entire limit is $0$

I pointed out where I think I went wrong, but I am not able to correct it. What should be the right procedure?