Understanding Limits: Targets and Approaches

Introduction

When you first learned about limits in calculus, you likely encountered two different-looking definitions that your textbook claimed were “equivalent.” One used sequences and indices, while the other used those mysterious $\delta$ and $\epsilon$ symbols. Perhaps you wondered: Why do we need two definitions for the same concept? Are they really the same?

This article explores the conceptual machinery behind limits by introducing two intuitive ideas: targets (where we want our function values to land) and approaches (how we get close to a point in the domain). We’ll see that these definitions are equivalent in $\mathbb{R}$, but for a subtle and beautiful reason that connects to the Archimedean principle and the countable structure of the natural numbers.

Prerequisites

To follow this article, you should be comfortable with:

• Basic real analysis: epsilon-delta definitions of limits, sequences, and convergence
• Set theory: adherent points, domains, codomains, and preimages
• Elementary topology (helpful but not essential): neighborhoods and the concept of “approaching” a point

If you’ve taken a first course in real analysis and have seen both the sequential and $\delta$-$\epsilon$ definitions of limits, you have all the background you need.

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A target is an interval in the codomain

A target represents an interval in the codomain of a function. For example, $(L - \epsilon, L + \epsilon)$ is a target. We want a corresponding approach in the domain whose image falls inside the target. But what exactly is an approach?

An approach is a way of getting close to a point in the domain

Abstractly, an approach is a strategy for getting arbitrarily close to a value $x_0$ in the domain such that the elements of the approach are eventually as close to $x_0$ as we want.

Every target has an approach whose image is contained in it

Let $f : X \to \mathbb{R}$, $E \subset X$, and let $x_0$ be an adherent point of $E$.

The Sequential Approach

Consider a sequence $(x_n)_{n\in\mathbb{N}}$ with $x_n \in E$ satisfying $x_n \to x_0$.

What’s the approach here? Our strategy is to index elements of the domain and place them into a convergent sequence. Recall that $\forall \delta > 0$, $\exists N \in \mathbb{N}$ such that $\forall n \geq N$:

$$|x_n - x_0| < \delta$$

The approach is the set $\{x_n : n \geq N\}$ for sufficiently large $N$. Since $\delta$ can be made arbitrarily small and all $x_n \in E$, this approach captures how we get close to $x_0$.

Definition (Sequential Limit): We say $\lim_{x \to x_0} f(x) = L$ if $\forall \epsilon > 0$, $\exists N \in \mathbb{N}$ such that $\forall n \geq N$:

$$|f(x_n) - L| < \epsilon$$

We have the approach $\{x_n : n \geq N\}$ and the target $(L - \epsilon, L + \epsilon)$. But notice how restrictive this strategy is: we capture elements of $E$ via a mapping $\mathbb{N} \to E$. This is a countable way of approaching $x_0$. Depending on the nature of $E$, this may not be enough to capture all valid approaches—certain limits might be missed.

The $\delta$-$\epsilon$ Approach

Here’s the alternative definition from your analysis class:

Definition ($\delta$-$\epsilon$ Limit): We say $\lim_{x \to x_0} f(x) = L$ if $\forall \epsilon > 0$, $\exists \delta > 0$ such that $\forall x \in E$ with $0 < |x - x_0| < \delta$:

$$|f(x) - L| < \epsilon$$

Observe the strategic difference: instead of relying on $\mathbb{N}$ to index our approach, we define the approach as all points in the punctured $\delta$-neighborhood of $x_0$:

$$\{x \in E : 0 < |x - x_0| < \delta\}$$

This bypasses the countability limitation. The $\delta$-$\epsilon$ definition is, in general, stronger than the sequential approach.

Why are they equivalent in $\mathbb{R}$?

Surprisingly, these definitions are equivalent in $\mathbb{R}$. Why? The key is the Archimedean principle: $\forall \delta > 0$, $\exists N \in \mathbb{N}$ such that:

$$\frac{1}{N} < \delta$$

Since $\frac{1}{n} \to 0$ as $n \to \infty$, our sequential approach past index $N$ satisfies:

$$|x_n - x_0| < \frac{1}{n} < \delta$$

Key Insight: For every $\delta$-neighborhood approach, we can construct a sequence $(x_n)$ where $x_n \in E$ and:

$$|x_n - x_0| < \frac{1}{n}$$

Thus, the sequential approach $\{x_n : n \geq N\}$ is contained in the $\delta$-neighborhood approach:

$$\{x_n : n \geq N\} \subset \{x \in E : 0 < |x - x_0| < \delta\}$$

Illustration

For a given $\epsilon > 0$, the target is $(L - \epsilon, L + \epsilon)$ in the codomain.

• $\delta$-$\epsilon$ approach: $\{x \in E : 0 < |x - x_0| < \delta\}$
• Sequential approach: $\{x_n : n \geq N\}$ where $|x_n - x_0| < \frac{1}{n} < \delta$

The images of both approaches lie in the target since $\forall x$ in either approach:

$$|f(x) - L| < \epsilon \implies f(x) \in (L - \epsilon, L + \epsilon)$$

First-countability

This equivalence holds in $\mathbb{R}$ because $\mathbb{R}$ is first-countable: every point has a countable neighborhood base (e.g., $\{B_{1/n}(x_0) : n \in \mathbb{N}\}$). In first-countable spaces, sequential and topological limits coincide. In spaces that are not first-countable, the $\delta$-$\epsilon$ (topological) definition is strictly more general than the sequential definition.