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''of a curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013610/a0136101.png" /> with an infinite branch''
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$#C+1 = 26 : ~/encyclopedia/old_files/data/A013/A.0103610 Asymptote
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A straight line the distance of which from the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013610/a0136102.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013610/a0136103.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013610/a0136104.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013610/a0136105.png" />) as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013610/a0136106.png" /> (from one side) is satisfied. An inclined asymptote, with equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013610/a0136107.png" />, exists if and only if the limits
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013610/a0136108.png" /></td> </tr></table>
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''of a curve  $  y = f(x) $
 +
with an infinite branch''
  
exist as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013610/a0136109.png" /> (or as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013610/a01361010.png" />).
+
A straight line the distance of which from the point  $  (x, f(x)) $
 +
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  $  x = a $,
 +
where  $  f(x) \rightarrow + \infty $(
 +
$  - \infty $)
 +
as $  x \rightarrow a $(
 +
from one side) is satisfied. An inclined asymptote, with equation  $  y = kx + l $,
 +
exists if and only if the limits
  
Similar formulas are also obtained for parametrized (unbounded) curves in general parametric representation. In polar coordinates an asymptote of a curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013610/a01361011.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013610/a01361012.png" />, with slope angle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013610/a01361013.png" />, is defined by the condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013610/a01361014.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013610/a01361015.png" />. The distance <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013610/a01361016.png" /> of this asymptote from the coordinate origin is calculated by the formula
+
$$
 +
k  =  \lim\limits 
 +
\frac{f (x) }{x}
 +
,\ \
 +
= \lim\limits  [ f (x) - kx ] ,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013610/a01361017.png" /></td> </tr></table>
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exist as  $  x \rightarrow + \infty $(
 +
or as  $  x \rightarrow - \infty $).
  
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013610/a01361018.png" /> has the asymptote <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013610/a01361019.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013610/a01361020.png" />, 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013610/a01361021.png" /> are given by the equations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013610/a01361022.png" />. An inclined asymptote yields a simple (linear with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013610/a01361023.png" />) asymptotic approximation of the function:
+
Similar formulas are also obtained for parametrized (unbounded) curves in general parametric representation. In polar coordinates an asymptote of a curve $  r = r ( \phi ) $,
 +
where  $  r>0 $,
 +
with slope angle  $  \alpha $,  
 +
is defined by the condition  $  r \rightarrow + \infty $
 +
as  $  \phi \rightarrow \alpha $.  
 +
The distance  $  p $
 +
of this asymptote from the coordinate origin is calculated by the formula
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013610/a01361024.png" /></td> </tr></table>
+
$$
 +
= \lim\limits  |t| r ( \alpha + t )
 +
\  \textrm{ as }  t \rightarrow +0  ( \textrm{ or  as  }  t \rightarrow -0).
 +
$$
  
as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013610/a01361025.png" /> (or as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013610/a01361026.png" />).
+
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  $  y = ( \sin  x  ^ {2} )/x $
 +
has the asymptote  $  y = 0 $
 +
as  $  x \rightarrow \pm \infty $,
 +
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  $  (x  ^ {2} /a ^ {2} ) - (y  ^ {2} /b  ^ {2} ) = 1 $
 +
are given by the equations  $  (x/a) \pm (y/b)= 0 $.  
 +
An inclined asymptote yields a simple (linear with respect to  $  x $)
 +
asymptotic approximation of the function:
 +
 
 +
$$
 +
f (x)  =  k x + l + o (1)
 +
$$
 +
 
 +
as  $  x \rightarrow + \infty $(
 +
or as $  x \rightarrow - \infty $).
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  P.K. Rashevskii,  "A course of differential geometry" , Moscow  (1956)  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  L.D. Kudryavtsev,  "Mathematical analysis" , Moscow  (1973)  (In Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  P.K. Rashevskii,  "A course of differential geometry" , Moscow  (1956)  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  L.D. Kudryavtsev,  "Mathematical analysis" , Moscow  (1973)  (In Russian)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A.V. Pogorelov,  "Differential geometry" , Noordhoff  (1959)  (Translated from Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A.V. Pogorelov,  "Differential geometry" , Noordhoff  (1959)  (Translated from Russian)</TD></TR></table>

Latest revision as of 18:48, 5 April 2020


of a curve $ y = f(x) $ with an infinite branch

A straight line the distance of which from the point $ (x, f(x)) $ 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 $ x = a $, where $ f(x) \rightarrow + \infty $( $ - \infty $) as $ x \rightarrow a $( from one side) is satisfied. An inclined asymptote, with equation $ y = kx + l $, exists if and only if the limits

$$ k = \lim\limits \frac{f (x) }{x} ,\ \ l = \lim\limits [ f (x) - kx ] , $$

exist as $ x \rightarrow + \infty $( or as $ x \rightarrow - \infty $).

Similar formulas are also obtained for parametrized (unbounded) curves in general parametric representation. In polar coordinates an asymptote of a curve $ r = r ( \phi ) $, where $ r>0 $, with slope angle $ \alpha $, is defined by the condition $ r \rightarrow + \infty $ as $ \phi \rightarrow \alpha $. The distance $ p $ of this asymptote from the coordinate origin is calculated by the formula

$$ p = \lim\limits |t| r ( \alpha + t ) \ \textrm{ as } t \rightarrow +0 ( \textrm{ or as } t \rightarrow -0). $$

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 $ y = ( \sin x ^ {2} )/x $ has the asymptote $ y = 0 $ as $ x \rightarrow \pm \infty $, 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 $ (x ^ {2} /a ^ {2} ) - (y ^ {2} /b ^ {2} ) = 1 $ are given by the equations $ (x/a) \pm (y/b)= 0 $. An inclined asymptote yields a simple (linear with respect to $ x $) asymptotic approximation of the function:

$$ f (x) = k x + l + o (1) $$

as $ x \rightarrow + \infty $( or as $ x \rightarrow - \infty $).

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)
How to Cite This Entry:
Asymptote. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Asymptote&oldid=13212
This article was adapted from an original article by L.P. Kuptsov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article