CalcNotes The Precise Definition of a Limit

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Section 2.3 The Precise Definition of a Limit

Definition 2.3.1. Limit, precise definition.

Let $f(x)$ be defined on an open interval around $c\text{,}$ except possible at $c$ itself. The limit of $f(x)$ as $c$ approaches $c$ is $L$, written

\begin{equation*} \lim_{x\rightarrow c} f(x) = L \end{equation*}

if, for every $\epsilon \gt 0\text{,}$ there is a $\delta \gt 0$ such that, for all $x\text{,}$ if $0 \lt |x-c| \lt \delta\text{,}$ then $|f(x)-L| \lt \epsilon\text{.}$

This definition is used to prove theorems, e.g., the rules in the previous section. We don't focus on it, however, in order to emphasize applications.