Asymptote

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)

References

[a1]	A.V. Pogorelov, "Differential geometry" , Noordhoff (1959) (Translated from Russian)