# 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)