Limit at Infinity¶

Definition¶

Let $X\subseteq \mathbb{R}$ and let $f\colon X\to \mathbb{R}$ be a function. Assume there exists $M_0\in\mathbb{R}$ such that $(M_0,+\infty)\subseteq X\text{.}$ We say that $f$ converges to $L\in\mathbb{R}$ as $x\to +\infty\text{,}$ and we write $\displaystyle \lim_{x\to +\infty} f(x)=L\text{,}$ if for every $\varepsilon>0$ there exists $M\ge M_0$ such that

x>M \, \implies\, |f(x)-L|<\varepsilon.

Similarly, assume there exists $M_0\in\mathbb{R}$ such that $(-\infty,M_0)\subseteq X\text{.}$ We say that $f$ converges to $L\in\mathbb{R}$ as $x\to -\infty\text{,}$ and we write $\displaystyle \lim_{x\to -\infty} f(x)=L\text{,}$ if for every $\varepsilon>0$ there exists $M\le M_0$ such that

x<M \, \implies\, |f(x)-L|<\varepsilon.

Interpretation¶

A limit at infinity describes how a function behaves as the input grows without bound: $f(x)\to L$ as $x\to+\infty$ means that $f(x)$ stays as close to $L$ as we like once $x$ is large enough. It formalizes long-term or asymptotic behaviour: the line $y=L$ is then a horizontal asymptote of the graph.

A damped oscillating function approaching a horizontal line at L, with a shaded band from L minus epsilon to L plus epsilon that the graph stays inside for all x beyond M.

The shaded band is the strip $|y-L|<\varepsilon$ to the right of the threshold $M$ , and the definition asks that the graph stays inside it for all $x>M$ $M$ . The damped oscillation keeps crossing the dashed line $y=L$ $L$ : $f$ need not approach $L$ monotonically or from one side, only the distance $|f(x)-L|$ has to become small. A narrower band usually needs a larger $M$ .

The bound $M$ takes over the role of $\delta$ in the limit at a point: $\varepsilon$ is given first, and $M$ may depend on it. The assumption $(M_0,+\infty)\subseteq X$ makes sure that every $x>M$ belongs to the domain, so the condition says something about $f$ for all large inputs.

The definition is the continuous counterpart of the limit of a sequence, with $x>M$ in place of $n>N$ $N$ . If $f(x)\to L$ as $x\to+\infty$ $+ \infty$ , then also $f(n)\to L$ as $n\to\infty$ $\infty$ . The converse fails: $\sin(\pi n)=0$ for every integer $n$ , but $\sin(\pi x)$ has no limit as $x\to+\infty$ $+ \infty$ . Limits at infinity return later, for example in improper integrals over unbounded intervals.

Key Results¶

Limits at infinity describe behaviour far from any finite region.
A function may have different limits at $+\infty$ and $-\infty\text{.}$
Limits at infinity may also be infinite: $\displaystyle\lim_{x\to+\infty}f(x)=+\infty$ means that for every $K\in\mathbb{R}$ there exists $M\ge M_0$ such that $x>M\implies f(x)>K\text{;}$ the cases with $-\infty$ are analogous.

Examples¶

$\displaystyle \lim_{x\to+\infty}\frac{1}{x}=0\text{.}$
$\displaystyle \lim_{x\to-\infty}x^2=+\infty\text{.}$

Prerequisites¶

Function

Sources¶

Math I: Calculus · Lecture 5

Aliases: limit of a function at infinity Keywords: function limit Symbols: delta, element of, epsilon, infinity, pi, real numbers, subset

Last updated: 20 October 2026 03:02