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

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

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

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

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

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

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