Limit at infinity

Definition

Let $D \subseteq \mathbb{R}$ be a set that is not bounded above, and let $f \colon D \to \mathbb{R}$ be a function. A real number $L$ is the limit of $f$ at $+\infty$ if for each $\varepsilon > 0$ there is $M$ with $|f(x) - L| < \varepsilon$ for all $x \in D$ with $x > M$. The notation is $\lim_{x \to \infty} f(x) = L$. The limit at $-\infty$ is defined in the same way, with $x < M$.

Remarks

A sequence is a function on $\mathbb{N}$. For sequences this definition gives the limit of a sequence. For example, $\lim_{x \to \infty} 1/x = 0$. Mathlib expresses $x \to \infty$ by the filter Filter.atTop.

Explanations

L1 human

The limit at infinity is the value that $f(x)$ approaches as $x$ grows without bound. The graph then gets close to a horizontal line $y = L$.