# asymptotes of graph of rational function

Let  $f(x)\,=\,\frac{P(x)}{Q(x)}$  be a fractional expression where $P(x)$ and $Q(x)$ are polynomials    with real coefficients such that their quotient can not be reduced (http://planetmath.org/Division) 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

 $f(x)\;=\;H(x)+\frac{R(x)}{Q(x)}$

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

 $\deg{R(x)}<\deg{Q(x)}$

The graph of the rational function  $f$ may have asymptotes:

1. 1.

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

2. 2.

If  $\deg{H(x)}<1$  (i.e. $0$  or  $-\infty$) then the graph has the horizontal asymptote  $y=H(x)$.

3. 3.

If  $\deg{H(x)}=1$  then the graph has the skew asymptote  $y=H(x)$.

Proof of 2 and 3.  We have  $\displaystyle f(x)\!-\!H(x)=\frac{R(x)}{Q(x)}\,\to 0$   as   $|x|\to\infty$.

Remark.  Here we use the convention that the degree of the zero polynomial  is  $-\infty$.

Title asymptotes of graph of rational function AsymptotesOfGraphOfRationalFunction 2013-03-22 15:09:34 2013-03-22 15:09:34 eshyvari (13396) eshyvari (13396) 10 eshyvari (13396) Result msc 51N99 msc 26C15 msc 26A09 Polytrope