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An asymptote to a curve is a straight line that the curve approaches in such a manner that it becomes as close as one might wish to the line by going far enough along the line. A "real-life" example of such an asymptotic relationship would involve a kitten standing 1 m from a box; and, if every hour the kitten walks halfway to the box; this results in a situation in which the kitten never reaches the box; because, the distance it travels, during each hour, is never more than halfway to the box. (Compare Zeno's paradoxes.)

A specific example of asymptotes can be found in the graph of the function f(x) = 1 / x, in which two asymptotes are being approached: the line y = 0 and the line x = 0. The curve approaches them, but, never reaches them. A curve approaching a vertical asymptote (such as in the preceding example's y = 0, which has an undefined slope) could be said to approach an " infinite limitA limit can be: Limit (mathematics), including: Limit of a function Limit of a sequence Net (topology) Limit (category theory) A constraint (mathematical, physical, economical, legal, etc. in the form of an inequality, such as: Chandrasekhar limit Greisen", although infinityInfinity is a word carrying a number of different meanings in mathematics, philosophy, theology and everyday life. In theology, for instance in the work of Duns Scotus, the infinity of God carries the sense not so much of quantity (leading to the question is not technically considered a limit. A curve approaching a horizontalHorizontal is an orientation relating to, or in parallel with the horizon, and the opposite of vertical. A horizontal line goes from left to right (or vice versa) where a vertical line goes from top to bottom. In Cartesian coordinates of the form x ''y , asymptote (such as in the preceding example's x = 0, which has a slope of 0) does approach a " true limitA limit can be: Limit (mathematics), including: Limit of a function Limit of a sequence Net (topology) Limit (category theory) A constraint (mathematical, physical, economical, legal, etc. in the form of an inequality, such as: Chandrasekhar limit Greisen".

Asymptotes do not need to be parallel to the x- and y-axes, as shown by the graph of x + x^-1, which is asymptotic to both the y-axis and the line y = x. When an asymptote is not parallel to the x or y axes, it is called an oblique asymptote.

A function f(x) can be said to be asymptotic to a function g(x) as x→∞. This has any of four distinct meanings:

1. f(x) - g(x) → 0.
2. f(x) / g(x) → 1.
3. f(x) / g(x) has a nonzero limit.
4. f(x) / g(x) is bounded and does not approach zero. See Big O notationIn complexity theory, computer science, and mathematics the Big O notation is a mathematical notation used to describe the asymptotic behavior of functions. More exactly, it is used to describe an asymptotic upper bound for the magnitude of a function in.