Vertical Asymptotes and Holes

Definition Let $f$ be defined on an open interval $(a, a + d)$ for some $d > 0$. The statement

means that for every number $K$ there is a number $\delta$ with $0 < \delta \le d$ such that $f(x) > K$ whenever $a < x < a + \delta$. The statement $f(x) \to -\infty$ as $x \to a^+$ means the same with $f(x) < K$ in place of $f(x) > K$. For the left side, written $x \to a^-$, use an interval $(a - d, a)$ and the inputs with $a - \delta < x < a$. In limit notation these statements read $\lim_{x \to a^+} f(x) = \infty$, $\lim_{x \to a^-} f(x) = -\infty$ and so on.

Definition The line $x = a$ is a vertical asymptote of the graph of $f$ if at least one of the following four statements is true.

For a rational function, $a$ is then an excluded value, so the graph never meets the line $x = a$.

Definition Let $r = \frac{p}{q}$ be a rational function with reduced form $\frac{P}{Q}$, so that $P$ and $Q$ share no factor of positive degree. If $Q(a) = 0$ for a real number $a$, then $P(a) \ne 0$, because otherwise $x - a$ would be a common factor. Near $a$ the numerator stays close to the nonzero number $P(a)$ while the denominator approaches $0$. The denominator has only finitely many zeros, so it keeps one sign just left of $a$ and one sign just right of $a$. Hence on each side of $a$ the values $r(x)$ tend to $\infty$ or to $-\infty$, and $x = a$ is a vertical asymptote. If $Q(a) \ne 0$, the values $r(x)$ approach the finite number $\frac{P(a)}{Q(a)}$ and there is no vertical asymptote at $a$. So the vertical asymptotes of $r$ are exactly the lines $x = a$ where $a$ is a real zero of the reduced denominator $Q$.

Definition With the same notation, let $a$ be excluded from the domain of $r$ while $Q(a) \ne 0$. Then the graph of $r$ has a removable hole at the point

The values of $r$ stay bounded near $a$ and approach the height of the hole from both sides. A vertical asymptote and a hole can never occur at the same $a$, since one needs $Q(a) = 0$ and the other needs $Q(a) \ne 0$.

Definition Let $x = a$ be a vertical asymptote of $r$ and write its reduced form as

where $m \ge 1$ is the multiplicity of $a$ as a zero of $Q$ and $S(a) \ne 0$. Polynomials have no jumps, so $P$ and $S$ keep the signs of $P(a)$ and $S(a)$ on a small interval around $a$. On that interval the sign of $r(x)$ is the sign of $\frac{P(a)}{S(a)}$ times the sign of $(x - a)^m$.

1. If $m$ is odd, the factor $(x - a)^m$ is negative to the left of $a$ and positive to the right, so the two sides go in opposite directions, one toward $\infty$ and the other toward $-\infty$.
2. If $m$ is even, the factor $(x - a)^m$ is positive on both sides, so both sides go toward $\infty$ when $\frac{P(a)}{S(a)} > 0$ and both go toward $-\infty$ when $\frac{P(a)}{S(a)} < 0$.

Definition The symbols $\infty$ and $-\infty$ do not stand for real numbers. The statement $f(x) \to \infty$ as $x \to a^+$ describes the values of $f$ at inputs near $a$ and says nothing about $f(a)$. At a vertical asymptote of a rational function, $a$ is not in the domain, so $f(a)$ is undefined, and writing $f(a) = \infty$ or $\frac{1}{0} = \infty$ is incorrect. A one-sided limit that equals $\infty$ or $-\infty$ does not exist as a real number, and the notation only records the way in which it fails to exist.