Limit at infinity

A limit at infinity for a function $f:D\to\mathbb R$ is a number $L\in\mathbb R$ such that: for every $\varepsilon>0$ there exists $M\in\mathbb R$ with the property that whenever $x\in D$ and $x>M$, one has $|f(x)-L|<\varepsilon$. One writes $\lim_{x\to\infty} f(x)=L$. Similarly, $\lim_{x\to-\infty} f(x)=L$ means that for every $\varepsilon>0$ there exists $M$ such that $x<M$ implies $|f(x)-L|<\varepsilon$.

This is an asymptotic version of the limit at a point definition, with “$x$ close to $a$” replaced by “$x$ large in magnitude.” It is commonly used to describe end behavior on unbounded intervals .

Examples:

• $\lim_{x\to\infty}\tfrac1x=0$.
• $\lim_{x\to\infty}\tfrac{2x+1}{x}=2$.