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 $\mathrm{\infty}$.
Title  asymptotes of graph of rational function 

Canonical name  AsymptotesOfGraphOfRationalFunction 
Date of creation  20130322 15:09:34 
Last modified on  20130322 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 