asymptotes of graph of rational function

Let  f(x)=P(x)Q(x)  be a fractional expression where P(x) and Q(x) are polynomialsMathworldPlanetmathPlanetmathPlanetmath with real coefficients such that their quotient can not be reduced ( to a polynomial.  We suppose that P(x) and Q(x) have no common zeros.

If the division of the polynomials is performed, then a result of the form


is gotten, where H(x) and R(x) are polynomials such that


The graph of the rational functionMathworldPlanetmath f may have asymptotes:

  1. 1.

    Every zero a of the denominator Q(x) gives a vertical asymptote  x=a.

  2. 2.

    If  degH(x)<1  (i.e. 0  or  -) then the graph has the horizontal asymptote  y=H(x).

  3. 3.

    If  degH(x)=1  then the graph has the skew asymptote  y=H(x).

Proof of 2 and 3.  We have  f(x)-H(x)=R(x)Q(x)0   as   |x|.

Remark.  Here we use the convention that the degree of the zero polynomialMathworldPlanetmath is  -.

Title asymptotes of graph of rational function
Canonical name AsymptotesOfGraphOfRationalFunction
Date of creation 2013-03-22 15:09:34
Last modified on 2013-03-22 15:09:34
Owner eshyvari (13396)
Last modified by eshyvari (13396)
Numerical id 10
Author eshyvari (13396)
Entry type Result
Classification msc 51N99
Classification msc 26C15
Classification msc 26A09
Related topic Polytrope