Horizontal Asymptotes

Definition Let $f$ be defined on an interval $(c, \infty)$. We write $\lim_{x \to \infty} f(x) = L$ if for every $\varepsilon > 0$ there is a number $M$ such that

The limit as $x \to -\infty$ is defined the same way on an interval $(-\infty, c)$, with $x < M$ in place of $x > M$. The line $y = L$ is a horizontal asymptote of the graph of $f$ if

Definition For a polynomial $p$ of degree $n$ with leading term $a_n x^n$,

Let $r = \frac{p}{q}$ be a rational function whose numerator has leading term $a_n x^n$ and whose denominator has leading term $b_m x^m$. For every $x \ne 0$ in the domain,

and the second factor tends to $1$ as $x \to \infty$ and as $x \to -\infty$. The leading terms are the dominant terms, and far out $r(x)$ behaves like $\frac{a_n}{b_m} x^{n - m}$, in the sense that the ratio of the two tends to $1$. In practice, divide the numerator and the denominator by $x^m$, the highest power of $x$ in the denominator, and use the fact that $\frac{c}{x^k} \to 0$ as $x \to \pm\infty$ for every constant $c$ and every integer $k \ge 1$.

Definition Let $r = \frac{p}{q}$, where the numerator $p$ is a nonzero polynomial of degree $n$ with leading coefficient $a_n$ and the denominator $q$ has degree $m$ and leading coefficient $b_m$.

1. If $n < m$, then $r(x) \to 0$ as $x \to \pm\infty$, and $y = 0$ is the horizontal asymptote.
2. If $n = m$, then $r(x) \to \frac{a_n}{b_m}$ as $x \to \pm\infty$, and $y = \frac{a_n}{b_m}$ is the horizontal asymptote.
3. If $n > m$, then $\lvert r(x) \rvert \to \infty$ as $x \to \pm\infty$, and there is no horizontal asymptote.

Cancelling a common factor lowers both degrees by the same amount and leaves the ratio of the leading coefficients unchanged, so the comparison gives the same answer before and after simplifying.

Definition The limits as $x \to \infty$ and as $x \to -\infty$ are two separate questions. A function may have a horizontal asymptote in one direction only, or a different one in each direction. Each limit has at most one value, so a graph has at most two horizontal asymptotes. For a rational function the degree comparison shows that if one of the two limits is a finite number $L$, the other one is $L$ as well, so a rational function has at most one horizontal asymptote $y = L$. Even then the graph may approach that line from above at one end and from below at the other. The sign of $r(x) - L$ for large positive $x$ and for large negative $x$ decides the side at each end.

Definition A horizontal asymptote describes $f(x)$ only as $x \to \infty$ or $x \to -\infty$, so nothing stops the graph from meeting the line $y = L$ at other places, even many times. For a rational function $r = \frac{p}{q}$ with horizontal asymptote $y = L$, the graph meets the line exactly at the points $(a, L)$ where $a$ is in the domain and

Unless $r$ is constant on its domain, the polynomial $p - Lq$ is not the zero polynomial and has degree less than $\deg q$, so there are at most $\deg q - 1$ such points. Vertical asymptotes are different, since the graph of a rational function never meets one.