# Can you have a horizontal and slant asymptote?

**You**may

**have**0 or 1

**slant asymptote**, but no more than that. A graph

**can have**both a vertical and a

**slant asymptote**, but it CANNOT

**have**both a

**horizontal and slant asymptote**.

**You**draw a

**slant asymptote**on the graph by putting a dashed

**horizontal**(left and right) line going through y = mx + b.

Likewise, can a slant asymptote be crossed?

NOTE: A common mistake that students make is to think that a graph cannot **cross** a **slant** or horizontal **asymptote**. This is not the case! A graph **CAN cross slant** and horizontal **asymptotes** (sometimes more than once). It's those vertical **asymptote** critters that a graph cannot **cross**.

**The three rules that horizontal asymptotes follow are based on the degree of the numerator, n, and the degree of the denominator, m.**

- If n < m, the horizontal asymptote is y = 0.
- If n = m, the horizontal asymptote is y = a/b.
- If n > m, there is no horizontal asymptote.

Consequently, what is the equation of the horizontal or oblique asymptote?

Case 1: If the degree of the numerator of f(x) is less than the degree of the denominator, i.e. f(x) is a proper rational function, the x-axis (y = 0) will be the **horizontal asymptote**. The line y = mx + b is an **oblique asymptote** for the graph of f(x), if f(x) gets close to mx + b as x gets really large or really small.

**Horizontal asymptotes** are **horizontal** lines that the graph of the function approaches as x tends to +∞ or −∞. As the name indicates they are parallel to the x-axis. Vertical **asymptotes** are vertical lines (perpendicular to the x-axis) near which the function grows without bound.