How are asymptotes actually defined in rigorous mathematics? This question is coming from the analytic geometry viewpoint. Please ignore the viewpoint of algebraic geometry here, unless that viewpoint is somehow able to handle non-algebraic curves like $x \mapsto e^x$ etc. which I don't think it is.
According to my Year 12 mathematics textbook, an asymptote is:

A line that approaches a curve but does not touch it.

With the benefit of the modern viewpoint, this definition just feels very weird and 18th Century, for the following reason: to "approach a curve" is to be somehow "close to it"; but, according to the above definition, if we get so close to the curve that we actually intersect it, then we're "too close" and its no longer an asymptote. Presumably, this isn't desirable, and so the hope is that modern authors no longer include this condition in their definition of "asymptote."
A moment of Googling brings up the relevant Wikipedia page:

In analytic geometry, an asymptote of a curve is a line such that the
  distance between the curve and the line approaches zero as they tend
  to infinity. Some sources include the requirement that the curve may not cross the line infinitely often, but this is unusual for modern authors.

Okay, this is definitely making a lot more sense. But, I still think a few things are screwy. Firstly, there are different definitions of curves lying about, and they aren't completely consistent. So lets just get rid of any mention of curves at all. We obtain:

Let $A$ denote a subset of $\mathbb{R}^2$. Then an asymptote of $A$ is a line such that the distance between $A$ and the line approaches zero as they tend to infinity.

Alright. Its still imprecise. Notice we haven't told the reader what exactly is approaching infinity. There's at least two ways of resolving this ambiguity, and they result in two fundamentally different notions of "asymptote."

Asymptote, Undirected Version. Let $A$ denote a subset of $\mathbb{R}^2$. Then an asymptote of $A$ is a line $L \subseteq \mathbb{R}^2$ such that for all $\varepsilon \in \mathbb{R}_{>0}$, there exists $r \in \mathbb{R}_{>0}$ such that for all $l \in L$, if $d(l,0)>r$, then the distance $d(l,A)$ is less than $\varepsilon$.

For example, the function $x \mapsto e^{-x^2}$ has precisely one asymptote according this definition. And the function $x \mapsto e^x$ has no asymptotes at all.
There's also another possible way of resolving the ambiguity. First, we need an auxiliary definition.

Directed Line. A directed line in $\mathbb{R}^2$ is a function $c : [0,\infty) \rightarrow \mathbb{R}^2$ such that:
  
  
*
  
*$c$ is injective
  
*$c$ preserves convex combinations: that is, for all $a,b \in [0,\infty)$ satisfying $a+b=1,$ we have $c(ap+bq) = ac(p)+bc(q).$
  
  
  Two directed lines are said to be equal iff they intersect at more than one point.

We can then define asymptote like so:

Asymptote, Directed Version. Let $A$ denote a subset of $\mathbb{R}^2$. Then an asymptote of $A$ is a directed line $c : [0,\infty) \rightarrow \mathbb{R}^2$ such that for all $\varepsilon \in \mathbb{R}_{>0}$, there exists $t \in \mathbb{R}_{\geq 0}$ such that for all $t' \geq t$, the distance $d(c(t'),A)$ is less than $\varepsilon$.

For example, the previously-mentioned function $f : \mathbb{R} \rightarrow \mathbb{R}$ given by $f(x)=e^{-x^2}$ has precisely two asymptotes according this definition. And the function $x \mapsto e^x$ has precisely one asymptote.
Now clearly, the undirected definition is easier to state. But I feel the the directed one is more useful. Anyway, my question is simple:

Question. How are asymptotes actually defined in rigorous mathematics?
I am not suggesting that there can be only one definition. If multiple definitions are useful, then I'd like to know about all of them. Thanks.

 A: How does the following work. Let $A\subseteq \Bbb R$ be an interval and $c: A \rightarrow \Bbb{R}^2$ continuous such that $c(A)$ is unbounded in the sense of the usual metric $d$ on $\Bbb R^2$. We can certainly find a sequence $a_n$ in $A$ such that for each $r>0,\,\exists N \in \Bbb N$  such that for $n>N$ we have $c(a_n) \not\in B_{r}((0,0))$. Now let $L\subset \Bbb R^2$ be a line, we say $L$ is an asymptote to $c$ if $\lim\limits_{n\rightarrow\infty} \mathrm{inf}\{d(c(a_n),l)\mid l\in L\}=0$, for a sequence $a_n$ as defined above.
Note you then add restrictions on how many times the curve is allowed to intersect a Line. Otherwise a Line is an asymptote to itself, and in fact any line is an asymptote to a spiral centered at the origin, eg Archimedean spiral. 
A: I think this may be more what you're interested in here. The author is Herbert Busemann, and the paper is on Local Metric Geometry, and he mentions asymptotes in metric spaces, where the distance is not necessarily symmetric, and according to the reference here on project Euclid, by Nasu, he says the concept of asymptotes was introduced. Check his references 2 and 3. That's one of them but I couldn't find the acta math one
