Divergence to Infinity¶

Definition¶

Let $f$ be a function defined on some punctured neighbourhood of $x_0\text{.}$ We say that $f$ diverges to positive infinity as $x\to x_0$ and write $\displaystyle \lim_{x\to x_0} f(x)=+\infty\text{,}$ if for every $M\in\mathbb{R}$ there exists $\delta>0$ such that

0<|x-x_0|<\delta \,\implies\, f(x)>M.

Similarly, we say that $f$ diverges to negative infinity as $x\to x_0$ and write $\displaystyle\lim_{x\to x_0} f(x)=-\infty\text{,}$ if for every $M\in\mathbb{R}$ there exists $\delta>0$ such that

0<|x-x_0|<\delta \,\implies\, f(x)<M.

Interpretation¶

Divergence to infinity describes functions whose values grow beyond all bounds near a given point. It is not a limit in the ordinary sense, since $+\infty$ is not a real number: the notation $\lim_{x\to x_0}f(x)=+\infty$ records the particular way in which the limit fails to exist.

A sequence growing with small wiggles, with a horizontal line at a level M and a vertical line at an index N after which all terms stay above M.

The figure shows the sequence version from the Key Results, with the index $n$ in the role of $x$ and $n\to\infty$ in place of $x\to x_0$ $x_{0}$ . For the level $M$ drawn, all terms from $N$ onwards lie in the shaded region above $M$ . A single early term above $M$ , here $a_7$ , is not enough, since $a_8$ and $a_9$ fall below $M$ again. The small wiggles do not matter, as long as such an $N$ exists for every level $M$ .

This modifies the definition of a limit at a point: instead of entering every small band about a finite value, the outputs must eventually exceed every real threshold, or fall below every threshold for negative divergence. The quantifiers work as in the $\varepsilon$ - $\delta$ definition: the level $M$ is given first and is arbitrary, and $\delta$ may depend on it, typically shrinking as $M$ grows. For $f(x)=1/|x|$ at $x_0=0$ $0$ , the choice $\delta=1/M$ works for every $M>0$ $0$ .

A common slip is to equate "unbounded" with "diverges to infinity". The sequence with $a_n=n$ for even $n$ and $a_n=0$ for odd $n$ is unbounded, yet it does not diverge to $+\infty$ , because infinitely many terms stay at $0$ . Divergence to infinity, possibly one-sided, is what defines a vertical asymptote.

Key Results¶

Divergence to infinity is a form of non-finite limit.
Divergence may differ on one-sided neighbourhoods.
A sequence $(a_n)_{n=1}^\infty$ diverges to $+\infty\text{,}$ written $a_n\to+\infty\text{,}$ if for every $M\in\mathbb{R}$ there exists $N\geq1$ such that $a_n>M$ for all $n\geq N\text{;}$ divergence to $-\infty$ is defined with $a_n<M\text{.}$

Examples¶

$\displaystyle \lim_{x\to 0}1/|x|=+\infty\text{.}$
$\displaystyle \lim_{x\to 0^-}1/x=-\infty\text{.}$

Prerequisites¶

Limit at a Point

Sources¶

Math I: Calculus · Lecture 5

Aliases: infinite limit Keywords: diverge to infinity, diverges to infinity Symbols: delta, element of, epsilon, infinity, real numbers

Last updated: 20 October 2026 03:02