Rigorous interpretation of $\lim_{\|x\|\rightarrow\infty}f(x)$ Let $D \subseteq \mathbb{R}^n$ and let $f:D\rightarrow \mathbb{R}$. In nonlinear stability theory, the idea of "radial unboundedness" see, e.g. this wikipedia article, is important. In most of the literature it is stated that a function is radially unbounded if, for $x\in D$,  "$\|x\|\rightarrow \infty \Rightarrow f(x) \rightarrow \infty$" or something similar, with a definition for the vague premise of the implication given as something similar to "$x$ is taken along any outgoing path / path that goes to infinity."
I am finding this definition too vague to be very useful for actual calculations. So I want to propose the following more precise definition:
Definition. Let $D$ be an unbounded subset of $\mathbb{R}^n$. A function $f:D\rightarrow \mathbb{R}$ is called radially unbounded for every real number $a > 0$ there exists another real number $r > 0$ such that 
$
\inf\{f(x) : \|x\| \geq r, \ x \in D\} > a.
$
My question is whether or not anyone else can see any discrepancies between this definition and what is implied by the imprecise definition. 
 A: The name radially unbounded sounds misleading to me, I would prefer having limit $+\infty$ at infinity, but never mind. 
Your definition 

For any $a > 0$ there is $r > 0$ such that $f(x) > a$ for all $x \in D$ with $\| x \| \ge r$.

seems all right; it corresponds to Cauchy's "$\varepsilon$-$\delta$" definition of a limit. The imprecise definition you're quoting can be made precise as follows: 

For any sequence $x_n \in D$ such that $\| x_n \| \to \infty$ we have $f(x_n) \to \infty$. 

This corresponds to Heine's definition of a limit, and it's equivalent to the previous one. 
Another (more topological) point of view would be to formalize limit at infinity by adding the infinity as an additional point in the domain; this procedure is known as Alexandrov's (or one-point) compactification. It enables us to consider $f$ as a function $f \colon D \cup \{ \infty \} \to \mathbb{R}$ (where $D \cup \{ \infty \}$ is a topological space). Then our definition is equivalent to 

The limit of $f$ at the point $\infty$ is $+\infty$ . 

