Vertical and horizontal asymptotes

Parametric equations

Let

After common factors are cancelled, if $x=a$ makes the denominator zero and the numerator nonzero, then $x=a$ is a vertical asymptote. As $x\to a^{\pm }$, $R(x)$ may go to $+\infty$ or to $-\infty$. If the numerator and the denominator are both zero at $x=a$, the point may be a removable discontinuity, a hole. For example, $\frac{x-1}{x-1}$ has no vertical asymptote at $x=1$.

A horizontal asymptote is read from the leading terms after cancellation. Let $\mathrm{deg}P=m$ and $\mathrm{deg}Q=n$. If $m<n$,

If $m=n$, the limit is the ratio of the leading coefficients, $y=\frac{a_m}{b_n}$. If $m>n$, there is no horizontal asymptote. When $m=n+1$, read an oblique asymptote instead. A larger gap is outside this page.

The factor control draws

Here the degrees match, so the horizontal asymptote is $y=A$. You move the zero $r$, the vertical asymptote $a$, and the far scale $A$. If $r$ meets $a$, the factor cancels: the figure drops the vertical asymptote and marks a hole.

Interaction

• Scale A, Zero r, and Asymptote a: move the zero, the denominator zero, and the far height together
• Factors / Hole / m<n / m=n / m>n: switch the family. m<n draws $R(x)=A/[(x-a)(x-h)]$, two vertical asymptotes and the horizontal asymptote $y=0$. m=n draws $R(x)=b+A/(x-a)$; Height b is the leading-coefficient ratio. m>n draws $R(x)=c(x-a)+b+1/(x-a)$. The gap is $1$, so there is no horizontal asymptote, and Slope c is the slope of the oblique line
• Hole and Hole marks: Hole draws $R(x)=A(x-r)(x-h)/[(x-a)(x-h)]$. The factor $x-h$ cancels and leaves a hole. Hole marks shows or hides it. Advanced opens Hole h / second root, which is also the second denominator zero on m<n
• Asymptotes: show or hide the vertical and horizontal guides. On m>n, Advanced draws the oblique line with those guides
• Local window: with Advanced on, magnify a denominator zero and see whether the two sides go the same way or opposite ways

What to notice

• A vertical asymptote is a denominator zero that remains after cancellation. A shared root can be a hole, not a vertical asymptote.
• A horizontal asymptote comes from the leading terms after cancellation. If $m<n$, the graph approaches $0$. If $m=n$, it approaches the ratio of the leading coefficients.
• If the numerator degree is higher, there is no horizontal asymptote. When the gap is $1$, read an oblique asymptote. A larger gap is outside this page.

Related

• Oblique asymptotes and polynomial division
• Polynomial zeros and multiplicity
• Rational functions and asymptotes

Further reading

• Asymptote (Wikipedia)
• Rational function (Wikipedia)