Limits and Continuity - Core Definitions

The concept of a limit is the foundation upon which all of calculus is built. Understanding limits allows us to rigorously define continuity, derivatives, and integrals, making them essential to mathematical analysis.

DefinitionLimit of a Function

Let $f$ be a function defined on an open interval containing $a$ , except possibly at $a$ itself. We say that the limit of $f(x)$ as $x$ approaches $a$ is $L$ , written as $\lim_{x \to a} f(x) = L$ if for every $\epsilon > 0$ , there exists a $\delta > 0$ such that whenever $0 < |x - a| < \delta$ , we have $|f(x) - L| < \epsilon$ .

This epsilon-delta definition, formalized by Weierstrass, captures the intuitive notion that we can make $f(x)$ arbitrarily close to $L$ by taking $x$ sufficiently close to $a$ . The condition $0 < |x - a|$ ensures that we never actually evaluate $f$ at $a$ itself, which is crucial since the limit may exist even when $f(a)$ is undefined.

DefinitionOne-Sided Limits

We define right-hand limit as $\lim_{x \to a^+} f(x) = L$ if for every $\epsilon > 0$ , there exists $\delta > 0$ such that $a < x < a + \delta$ implies $|f(x) - L| < \epsilon$ .

Similarly, the left-hand limit is $\lim_{x \to a^-} f(x) = L$ if for every $\epsilon > 0$ , there exists $\delta > 0$ such that $a - \delta < x < a$ implies $|f(x) - L| < \epsilon$ .

Remark

A limit $\lim_{x \to a} f(x) = L$ exists if and only if both one-sided limits exist and are equal: $\lim_{x \to a^-} f(x) = \lim_{x \to a^+} f(x) = L$ This characterization is often useful for analyzing piecewise functions.

ExampleComputing a Simple Limit

Consider $\lim_{x \to 2} (3x - 1)$ . Intuitively, as $x$ approaches 2, $3x - 1$ approaches $3(2) - 1 = 5$ . To prove this rigorously, let $\epsilon > 0$ be given. We need to find $\delta > 0$ such that $0 < |x - 2| < \delta \implies |3x - 1 - 5| < \epsilon$

Simplifying: $|3x - 6| = 3|x - 2| < \epsilon$ , so $|x - 2| < \epsilon/3$ . Therefore, choosing $\delta = \epsilon/3$ works, and we conclude $\lim_{x \to 2} (3x - 1) = 5$ .

Understanding these foundational definitions is essential for developing the limit laws and continuity properties that follow.