Limits at Infinity and Negative Infinity

This page is intended to be a part of the Real Analysis section of Math Online. Similar topics can also be found in the Calculus section of the site.

 Definition: Let $f : A \to \mathbb{R}$ be a function. We say that the limit of $f$ at $\infty$ is $L$ written $\lim_{x \to \infty} f(x) = L$ if $\forall \epsilon > 0$ $\exists M > 0$ such that if $x \in A$ and $x > M$ then $\mid f(x) - L \mid < \epsilon$. Similarly, we say that the limit of $f$ at $-\infty$ is $L$ written $\lim_{x \to \infty} f(x) = L$ if $\forall \epsilon > 0$ $\exists M < 0$ such that if $x \in A$ and $x < M$ then $\mid f(x) - L \mid < \epsilon$.
The following diagram illustrates the concept of a limit at positive infinity. Given any $\epsilon > 0$, we can find a real number $M > 0$ such that all values of $x > M$ ($x \in A$) are within $\epsilon$ from $L$.
Similarly, given $\epsilon > 0$, we can find a real number $M < 0$ such that all values of $x < M$ ($x \in A$) are within $\epsilon$ from $L$ as illustrated in the following diagram: