Is enough to evaluate the limits at the edges of the domain to determine the range of $y = 5/(x^3+2)$? Is enough to evaluate the limits at the edges of the domain in order to determine the range of $y = 5/(x^3+2)$?
The domain is $(-\infty, -2^{1/3}) \cup (-2^{1/3}, +\infty)$ and calculating the limits at the edges of the domain, I obtain: $(-\infty,0) \cup (0,+\infty)$, which is the range. But I've checked it in a calculator and would like to understand if in this case I need more information in order to state the range. For instance, taking the derivative (I've done it) ed evaluate if/where the function decreases/increases… However, would like to know more from you, concerning this specific case and in general!

Your function is decreasing over $(-\infty, -2^{1/3})$ and $(-2^{1/3}, +\infty)$, because $y'(x)\le0$. It is then enough to evaluate the limits at the endpoints of the domain to get its range.
In general, if $f$ is not monotonic, you also have to find its maxima and minima over every interval of continuity.
