2.11. Oblique Asymptotes http://www.ck12.org
2.11 Oblique Asymptotes
Here you will extend your knowledge of horizontal and vertical asymptotes and learn to identify oblique (slanted)
asymptotes. You will also be able to apply your knowledge of polynomial long division.
When the degree of the numerator of a rational function exceeds the degree of the denominator by one then the
function has oblique asymptotes. In order to find these asymptotes, you need to use polynomial long division
and the non-remainder portion of the function becomes the oblique asymptote. A natural question to ask is: what
happens when the degree of the numerator exceeds that of the denominator by more than one?
Watch This
MEDIA
Click image to the left for use the URL below.
URL: http://www.ck12.org/flx/render/embeddedobject/60901http://www.youtube.com/watch?v=W8ASTRfEMVo James Sousa: Determining Slant Asymptotes of Rational Func-
tions
Guidance
The following function is shown before and after polynomial long division is performed.
f(x) =x^4 +^3 xx (^3) −^2 + 3 x^22 x+^14 =x+ 3 +^12 xx^23 +−^23 xx+ 214
Notice that the remainder portion will go to zero whenxgets extremely large or extremely small because the power
of the numerator is smaller than the power of the denominator. This means that while this function might go haywire
with small absolute values ofx, large absolute values ofxare extremely close to the liney=x+3.