PlanetMath (more info)
 Math for the people, by the people.
Encyclopedia | Requests | Forums | Docs | Wiki | Random | RSS  
Login
create new user
name:
pass:
forget your password?
Main Menu
Owner confidence rating: Very low Entry average rating: No information on entry rating
[parent] asymptotes of graph of rational function (Definition)

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 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

$\displaystyle f(x) = H(x)+\frac{R(x)}{Q(x)}$
is gotten, where $ H(x)$ and $ R(x)$ are polynomials such that
$\displaystyle \deg{R(x)} < \deg{Q(x)}$

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 $ \deg{H(x)} < 1$ (i.e. 0 or $ -\infty$) then the graph has the horizontal asymptote $ y = H(x)$.
  3. If $ \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) = \frac{R(x)}{Q(x)}\,\to 0$ as $ \vert x\vert\to\infty$.

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



"asymptotes of graph of rational function" is owned by eshyvari. [ owner history (1) ]
(view preamble)

View style:


This object's parent.
Log in to rate this entry.
(view current ratings)

Cross-references: zero polynomial, degree, proof, denominator, asymptotes, rational function, graph, division, quotient, coefficients, real, polynomials, fractional expression
There is 1 reference to this entry.

This is version 6 of asymptotes of graph of rational function, born on 2005-03-27, modified 2007-01-26.
Object id is 6908, canonical name is AsymptotesOfGraphOfRationalFunction.
Accessed 2733 times total.

Classification:
AMS MSC26A09 (Real functions :: Functions of one variable :: Elementary functions)
 26C15 (Real functions :: Polynomials, rational functions :: Rational functions)
 51N99 (Geometry :: Analytic and descriptive geometry :: Miscellaneous)

Pending Errata and Addenda
None.
Discussion
Style: Expand: Order:
forum policy

No messages.

Interact
post | correct | update request | add derivation | add example | add (any)