Definition. Let $A$ be a nonempty subset of $\mathbb{R}$. $x \in \mathbb{R}$ is called an almost upper bound of $A$ if there are only finitely many $y \in A$ for which $y \geq x$. Similarly we define almost lower bounds. Define $\lim \sup{A}$ to be the infimum of all almost upper bounds of $A$ and $\lim \inf{A}$ to be the supremum of all almost lower bounds of $A$.
What are $\lim \inf{\mathbb{Z}^+}$ and $\lim \sup{\mathbb{Z}^+}$ if $A$ is finite?
Since $\mathbb{Z}^+$ is of infinite size, I don't see how about the upper bound part since there will always be infinitely many greater than any $x$. Thus, how would we define it for the empty set?
Since $\mathbb{Z}^+$ is unbounded above, it has no almost upper bounds. So $\limsup\mathbb{Z}^+=\inf(\varnothing)$. This is either undefined, or equal to $\infty$ (depending on your convention).
Now note that every positive number is an almost lower bound of $\mathbb{Z}^+$, so $\liminf\mathbb{Z}^+=\sup[0,\infty)$, which is again either undefined or equal to $\infty$, depending on your convention.