Asymptote
of a curve 
with an infinite branch
A straight line the distance of which from the point 
on the curve tends to zero as the point moves along the branch of the curve to infinity. An asymptote can be vertical or inclined. The equation of a vertical asymptote is 
, where 
(
) as 
(from one side) is satisfied. An inclined asymptote, with equation 
, exists if and only if the limits
 |
exist as 
(or as 
).
Similar formulas are also obtained for parametrized (unbounded) curves in general parametric representation. In polar coordinates an asymptote of a curve 
, where 
, with slope angle 
, is defined by the condition 
as 
. The distance 
of this asymptote from the coordinate origin is calculated by the formula
 |
If there exists a limit position of the tangent line to the infinite branch of the curve, this position is an asymptote. The converse is not always true. Thus, the curve 
has the asymptote 
as 
, even though a limit position of the tangent line does not exist. Hyperbolas are the only second-order curves with asymptotes. The asymptotes of the hyperbola 
are given by the equations 
. An inclined asymptote yields a simple (linear with respect to 
) asymptotic approximation of the function:
 |
as 
(or as 
).
References
| [1] | P.K. Rashevskii, "A course of differential geometry" , Moscow (1956) (In Russian) |
| [2] | L.D. Kudryavtsev, "Mathematical analysis" , Moscow (1973) (In Russian) |
Comments
References
| [a1] | A.V. Pogorelov, "Differential geometry" , Noordhoff (1959) (Translated from Russian) |
Asymptote. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Asymptote&oldid=45234