Definition Of Asymptotes
More generally, one curve is a curvilinear asymptote of another (as opposed to a linear asymptote) if the distance between the two curves tends to zero as they tend to infinity, although the term asymptote by itself is usually reserved for linear asymptotes.
Definition of Asymptotes
Asymptotes convey information about the behavior of curves in the large, and determining the asymptotes of a function is an important step in sketching its graph. The study of asymptotes of functions, construed in a broad sense, forms a part of the subject of asymptotic analysis.
The degree of the numerator and degree of the denominator determine whether or not there are any horizontal or oblique asymptotes. The cases are tabulated below, where deg(numerator) is the degree of the numerator, and deg(denominator) is the degree of the denominator.
The vertical asymptotes occur only when the denominator is zero (If both the numerator and denominator are zero, the multiplicities of the zero are compared). For example, the following function has vertical asymptotes at x = 0, and x = 1, but not at x = 2.
Although the definition here uses a parameterization of the curve, the notion of asymptote does not depend on the parameterization. In fact, if the equation of the line is a x + b y + c = 0 \displaystyle ax+by+c=0 then the distance from the point A(t) = (x(t),y(t)) to the line is given by
An asymptote can be either vertical or non-vertical (oblique or horizontal). In the first case its equation is x = c, for some real number c. The non-vertical case has equation y = mx + n, where m and n \displaystyle n are real numbers. All three types of asymptotes can be present at the same time in specific examples. Unlike asymptotes for curves that are graphs of functions, a general curve may have more than two non-vertical asymptotes, and may cross its vertical asymptotes more than once.
Asymptotes are used in procedures of curve sketching. An asymptote serves as a guide line to show the behavior of the curve towards infinity. In order to get better approximations of the curve, curvilinear asymptotes have also been used  although the term asymptotic curve seems to be preferred.
The asymptotes of an algebraic curve in the affine plane are the lines that are tangent to the projectivized curve through a point at infinity. For example, one may identify the asymptotes to the unit hyperbola in this manner. Asymptotes are often considered only for real curves, although they also make sense when defined in this way for curves over an arbitrary field.
The curves visit these asymptotes but never overtake them. The method opted to find the horizontal asymptote changes involves comparing the degrees of the polynomials in the numerator and denominator of the function. If both the polynomials have the same degree, divide the coefficients of the largest degree terms.
A vertical asymptote occurs when x can't equal some value. For example, if you graph y = 1/x, you will see that x can't ever equal zero. So, there is a vertical asymptote at x = 0. There is also a horizontal asymptote at y = 0. A graph can have both kinds of asymptotes.
Hence, the two vertical asymptotes are x = 1 and x = -5. In fact, the value of y may go to positive or negative infinity as x approaches 1 or -5 along these vertical asymptotes.
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."
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:
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
I think what confuses is the fact that the definition in the book is just an informal way of saying what you said (modulo the line meeting the curve in 2 points - I'll get to that immediatly): In my expreience a lot of calculus books are deliberately imprecise so as not to frighten the student with technical definition and thus give first a definition which is supposed to appeal to the intuition. We could state the definition of an asymptote at different levels which vary in rigor and formalism. Thus we could have:
Less precise (at an essential point) and informal - your definition: "The asymptote to a curve is a straight line such that the distance between the curve and the line tends to zero as they tend to infinity" (Lack of precision concerning whether you meant only $+\infty$ or not and if not, if you meant "they tend to $+\infty$ or $-\infty$" or they tend to $ +\infty$ and $ -\infty$")
Less precise (at a not so essential point) and informal - The books definition: "An asymptote is a straight line which meets the curve at two coincident points at infinity" (Lack of precision concerning how to rigorously translate "meets the curve" into a mathematical statement)
More precise and semi-formal - Wikipedia's definition. Note that to be formal, one has to distinguish many cases in which the graph of the function could behave. Also note that in that definition arbitrary curves are excluded, since these can be rather monstruous and to ask for an asymptote for such a thing wouldn't make sense; see for example this thing ].
Note further that 1) the more formal you get, the more context you have to specify (which in the other cases is implicitly assumed. Example: You were only talking about curves of graphs of functions on arbitrary curves in $\mathbbR^2$) 2) these 3 levels aren't by far the only ones; one could insert many levels of rigor/formalism between these three and there would also by room. 3) definitions aren't carved in stone; different authors use slightly different definitions (which mostly vary in technicalities; the underlying intuition is almost always the same).
To address the last problem (of the first question) concerning the line meeting the curve in 2 points: This is a special case of the note 3) from above: Some authors state their one definition for various reasons. My guess, as to why the author required that the line should meet the curve in 2 points at infinity, is, again, that he wants to remain intuitive - and this definition of an asymptote just does that. The graph of the other answer illustrates this: the line given by all the point in $\mathbbR^2$ that satisfy the equation $y=0$ is, in his definition, an asymptote, which would also be clear by just looking at the graph (which accounts for the intuitiveness of the definition). Usually asymptotes are defined like in your definition, where one is only concerned if the line is "close enough" to the function graph either at $t \rightarrow +\infty$ (I have namend the variable of your function $t$) or $t \rightarrow -\infty$ (instead of $t \rightarrow +\infty$ and $t \rightarrow -\infty$ like in the books version), but then you would have examples of functions and lines, where the line is an asymptote to the function, but the resulting picture isn't so nice anymore, since the line would approximate" the function nicely only for, say, $t\rightarrow +\infty$.
After all this it, you should be now aware, that your definition is a little bit to vague , since you haven't specified if you meant with "infinity" only the positive infinity. If you did, the answer is no.
After reading all the answers and searching for a bit on the web I think I have found on the correct solution. I am grateful to everyone who responded. I don't think both definitions are mathematically formal. The first conveys the idea that the asymptote is a line which kisses the curve at infinity or in other words is a tangent to the curve at infinity.
The second "definition" essentially says that the tangent to the curve at infinity is of multiplicity two (The definition is silent on whether it may be more). I found this result (Proposition 6.0.3) on the internet which says that with a proper understanding of "multiplicity" at an intersection point the tangent and the curve always have multiplicity at least two. So the second statement conveys the same idea with the addition of multiplicity at the intersection.
The graph of a function can never cross the VA and hence it is NOT a part of the curve anymore. We find vertical asymptotes while graphing but it is not mandatory to show them on the graph. Even the graphing calculators do not show them explicitly with dotted lines.
Note that do not set the denominator = 0 directly without simplifying the function. If we do that, we get x = -1 and x = 1 to be the VAs of f(x) in the above example. But x = -1 is NOT a VA anymore in this case, because (x + 1) has got canceled while simplification. In fact, there will be a hole at x = -1. Here is the graph of the function to understand the difference between the vertical asymptotes and holes.
Among the 6 trigonometric functions, 2 functions (sine and cosine) do NOT have any vertical asymptotes. But each of the other 4 trigonometric functions (tan, csc, sec, cot) have vertical asymptotes. To find them, just think about what values of x make the function undefined. Here are the vertical asymptotes of trigonometric functions: 041b061a72