http://www.ck12.org Chapter 2. Polynomials and Rational Functions
Oblique asymptotesare these slanted asymptotes that show exactly how a function increases or decreases without
bound.
Example A
Identify the oblique asymptotes of the following rational function.
f(x) =xx^32 −+ 3 x−x−^334 =x− 3 +(x−^121 x)(−x^45 + 4 )
Solution: Since this function has been rewritten after long division has been performed, the oblique asymptote is
the line that remains:
y=x− 3
Example B
Identify the vertical and oblique asymptotes of the following rational function.
f(x) =x(x^3 −− 3 x)(^2 −x+x− 41 )
Solution: After using polynomial long division and rewriting the function with a remainder and non-remainder
portion it looks like this:
f(x) =x− 2 +x^132 +xx−−^2512 =x− 2 +(x−^133 x)(−x^25 + 4 )
The oblique asymptote isy=x−2. The vertical asymptotes are atx=3 andx=−4 which are easier to observe in
last form of the function because they clearly don’t cancel to become holes.
Example C
Identify the oblique asymptotes of the following rational function.
f(x) =(x^210 −(^4 x)(−x 1 +)^3 )
Solution: The degree of the numerator is 3 so the slant asymptote will not be a line. However when the graph is
observed, there is still a clear pattern as to how this function increases without bound asxapproaches very large and
very small numbers.
f(x) = 101 (x^2 + 4 x)− 10 (^12 x− 1 )
As you can see, this looks like a parabola with a remainder. This rational function has a parabola backbone. This
is not technically an oblique asymptote because it is not a line.
Concept Problem Revisited
When the numerator exceeds the denominator by more than one, the function develops a backbone as in Example C
that can be shaped like any polynomial. Oblique asymptotes are always lines.
Vocabulary
Oblique asymptotesare asymptotes that occur at a slant. They are always lines.