A slant asymptote will exist for **the rational function**. The slant asymptote is the quotient portion of the result obtained by dividing the numerator by the denominator. A graph can have both a vertical and a slant asymptote, but not both. Vertical asymptotes occur when the graph approaches a constant value as x gets large or small. Slant asymptotes occur when the graph approaches a constant value as y gets large or small.

- Can a rational function have both a horizontal and an oblique asymptote?
- Does the graph of a rational function always have a vertical asymptote?
- Can a rational graph ever cross an asymptote?
- Why can functions cross horizontal asymptotes?
- What are linear asymptotes?
- Why is it impossible for the graph of a rational function to cross the vertical asymptote?
- What are the asymptotes of rational functions?
- What are the rules for asymptotes?

A given rational function may or may not have a vertical asymptote (depending on whether the denominator ever equals zero), but it will always have either a horizontal or **a slant asymptote** (at this level of study). If the function has a horizontal asymptote, then it must approach infinity at some point to the right of the axis. If it has a slant asymptote, then it can approach either positive or negative infinity depending on which direction is up on the axis.

Here are some examples of functions with various types of asymptotes:

Any rational function has either a horizontal or a slant asymptote- any function that approaches infinity at some point to the right of the axis has a horizontal asymptote- any function that approaches minus infinity from below has a slant asymptote.

Functions with **no asymptotes** include exponents (e^x) and logarithms (log_b x).

A rational function's graph will never cross a vertical asymptote, but it may or may not cross **a horizontal or slant asymptote**. In addition, whereas the graph of a rational function may have multiple vertical asymptotes, it will only have one horizontal (or slant) asymptote. A picture is worth a thousand words: here are some graphs for functions defined by rational expressions.

Vertical A rational function has a vertical asymptote when its denominator is equal to zero. This does not apply to horizontal and oblique asymptotes. Horizontal Horizontal asymptotes provide information on the graph's extremes, or extrema, +-. As a result, graphs can cross a horizontal asymptote. Oblique Asymptotes are at an angle other than 90 degrees to the axis of the coordinate system. They provide information about the relationship between **two variables** that are not on the same scale.

Linear asymptotes can be found in graphs of rational functions. Vertical, horizontal, or slanted asymptotes are possible (also called oblique). Graphs can have **several types** of asymptotes. The methods below show how to locate the asymptote of a Rational Function . (s).

For example, if you have the function y = 1x2-1, you can get the vertical asymptote by setting the denominator to zero. But because the denominator of a rational function is never zero, the graph cannot cross **a vertical asymptote**.

When the degree of the denominator equals the degree of the numerator, a rational function has **a horizontal asymptote**. The degree is just the most powerful phrase. There is a subgroup of **horizontal asymptotes** that is unique. This occurs when the numerator's degree is smaller than the denominator's degree. If this is the case then the function will never reach **its horizontal asymptote** and will always be bounded below by a value less than or equal to zero.

If the degree of the denominator is greater than the degree of the numerator, then the function will always reach its horizontal asymptote and will always be bounded above by a value greater than or equal to zero.

In both cases, if you were to divide the function by its highest power of x (the degree of the denominator), then it would become a constant value. This means that there is no limit to how far down or up the function can go before it stops changing direction and starts going in a fixed direction.

For example, if we take the function $f(x) = \frac{1}{x}$, then $f$ has a horizontal asymptote at $x=0$. Because the degree of the numerator is smaller than the degree of the denominator, $f$ does not reach **its horizontal asymptote** and is therefore not bounded below by a value less than or equal to zero.

A rational function's horizontal asymptote can be found by examining the degrees of the numerator and denominator.

- Degree of numerator is less than degree of denominator: horizontal asymptote at y = 0.
- Degree of numerator is greater than degree of denominator by one: no horizontal asymptote; slant asymptote.