 # True or False. The graph of a rational function may intersect a horizontal asymptote. This article aims to determine whether the given statement is true or false. The statement is, “The graph of a rational function may intersect a horizontal asymptote.” This article uses the concept of horizontal asymptote of the rational function.

A horizontal asymptote is horizontal line that is not part of the graph of a function but leads it for $x$ values “far” right and “far” left. The graph may intersect it, but eventually, for large enough or small enough values of $x$, graph would get closer and closer to asymptote without touching it. Horizontal asymptote is a special case of an oblique asymptote.

Horizontal asymptote of rational function can be find by looking at degrees of the numerator and denominator.

If $N$ is the degree in the numerator and $D,$ is the degree in the denominator.

-$N < D$, then the horizontal asymptote is $y = 0$.

-$N = D$, then the horizontal asymptote is $y = ratio\: of\: leading\: coefficients$.

-$N > D$, then there is no horizontal asymptote.

The statement is true. It is possible that graph of a rational function can cross a horizontal asymptote.

Horizontal asymptote of a rational function can find by observing at the degrees of the numerator and denominator.

-The degree of the numerator is less than the degree of the denominator: horizontal asymptote at

-$y = 0$

-The degree of the numerator is greater than the degree of the denominator by one: no horizontal asymptote; oblique asymptote.

-The degree of the numerator is equal to the degree of the denominator: the horizontal asymptote in the ratio of the leading coefficients.

## Numerical Result

The statement is true. It is possible that the graph of a rational function can cross a horizontal asymptote.

## Example

True or False: Graph of a rational function $R$ never crosses a vertical asymptote. True or False: Graph of a rational function $R$ never crosses a horizontal asymptote. True or False: Graph of a rational function $R$ never crosses an oblique asymptote.

Solution

All statements are true.

An asymptote is a line along which the values of a function approaches but never reach, such that one or both of the $x$ or $y$ coordinates tend to positive or negative infinity. Therefore, the graph of a rational function $R$ never intersects any of its asymptotes.