Oblique asymptotes and polynomial division

Parametric equations

Degrees and the division below are for the rational function after common factors are cancelled. When $\mathrm{deg}P=\mathrm{deg}Q+1$, polynomial division

gives a linear quotient $S(x)$ and a remainder. Set

Because

the graph approaches the oblique asymptote $y=S(x)$ far away.

The figure uses

so $y=mx+b$ is the oblique asymptote, $A$ sets how far the graph leaves that line, and $x=c$ is the uncancelled denominator zero, a vertical asymptote.

For example,

approaches $y=x$ far away, and after cancellation $x=0$ is a vertical asymptote.

Interaction

• Slope m and Intercept b: move the quotient $S(x)=mx+b$ and watch the far line move
• Remainder strength A and Denominator zero c: change the remainder and the vertical asymptote, and compare the graph near $x=c$ with the line far away
• Asymptotes: show or hide the oblique line and the vertical line
• Advanced: show the decay of the remainder $E(x)=R(x)-S(x)$ with Remainder E, and compare degrees. Oblique is the line above. Horizontal draws $R(x)=b+A/(x-c)$, so the far line is the horizontal asymptote $y=b$. Toward 0 draws $R(x)=A/[(x-c)(x-d)]$, with horizontal asymptote $y=0$; Second zero d is the second denominator zero. Reset returns to the oblique parameters

What to notice

• The oblique asymptote is the quotient $S(x)$ from division after cancellation. The remainder has lower degree, and it sets how fast the graph settles onto the line.
• Compare degrees after cancellation. $\mathrm{deg}P=\mathrm{deg}Q+1$ is the usual condition for an oblique asymptote.
• The same rational function can have a vertical asymptote (an uncancelled denominator zero) and an oblique asymptote (how it behaves far away). They describe different scales.

Related

• Vertical and horizontal asymptotes
• Polynomial zeros and multiplicity
• Rational functions and asymptotes

Further reading

• Polynomial long division (Wikipedia)
• Asymptote (Wikipedia)