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 for every sequence $(x_n)_{n\in\mathbb{N}}$ with $x_n \in E$ and $x_n \to x_0$, we have $f(x_n) \to L$. That is, $\forall \epsilon > 0$, $\exists N \in \mathbb{N}$ 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 $\mathbb{R}$. The key is understanding how the countable sequential approach can capture the uncountable $\delta$-neighborhood approach.

From $\delta$-$\epsilon$ to Sequential

Given: The $\delta$-$\epsilon$ definition holds.

To show: Any sequence $(x_n)$ in $E$ converging to $x_0$ has $f(x_n) \to L$.

Proof: Let $(x_n)$ be any sequence in $E$ with $x_n \to x_0$. For any $\epsilon > 0$, the $\delta$-$\epsilon$ definition gives us some $\delta > 0$ such that all points in $E$ within distance $\delta$ of $x_0$ map to within $\epsilon$ of $L$. Since $x_n \to x_0$, there exists $N$ such that for all $n \geq N$, we have $|x_n - x_0| < \delta$. Therefore $|f(x_n) - L| < \epsilon$ for all $n \geq N$. ✓

From Sequential to $\delta$-$\epsilon$ (The Subtle Direction)

Given: The sequential definition holds (i.e., for every sequence in $E$ converging to $x_0$, the image sequence converges to $L$).

To show: The $\delta$-$\epsilon$ definition holds.

This is where the Archimedean principle becomes crucial: $\forall \delta > 0$, $\exists N \in \mathbb{N}$ such that:

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

Proof by construction: We prove this by contradiction. Suppose the $\delta$-$\epsilon$ definition fails. Then there exists some $\epsilon_0 > 0$ such that for every $\delta > 0$, there exists a point $x \in E$ with $0 < |x - x_0| < \delta$ but $|f(x) - L| \geq \epsilon_0$.

Now, construct a specific sequence as follows: For each $k \in \mathbb{N}$, take $\delta_k = \frac{1}{k}$. By our assumption, there exists $x_k \in E$ such that:

$$0 < |x_k - x_0| < \frac{1}{k} \quad \text{and} \quad |f(x_k) - L| \geq \epsilon_0$$

The sequence $(x_k)_{k\in\mathbb{N}}$ clearly converges to $x_0$ (since $|x_k - x_0| < \frac{1}{k} \to 0$), yet $f(x_k)$ does not converge to $L$ (since $|f(x_k) - L| \geq \epsilon_0$ for all $k$). This contradicts the sequential definition. Therefore, the $\delta$-$\epsilon$ definition must hold. ✓

The Role of the Archimedean Principle

The Archimedean principle ensures that the sequence $\frac{1}{k}$ can approximate any positive real number $\delta$ from below. This allows us to construct a sequence that “samples” arbitrarily close to $x_0$ in a systematic way.

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

$$|x_k - x_0| < \frac{1}{k}$$

When $k$ is large enough that $\frac{1}{k} < \delta$, the tail of this sequence $\{x_k : k \geq K\}$ is contained in the $\delta$-neighborhood:

$$\{x_k : k \geq K\} \subset \{x \in E : 0 < |x - x_0| < \delta\}$$

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: A constructed sequence $(x_k)$ where $|x_k - x_0| < \frac{1}{k}$

For sufficiently large $k$ (specifically, $k \geq K$ where $\frac{1}{K} < \delta$), the elements $x_k$ lie in the $\delta$-neighborhood, so their images satisfy:

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

First-countability

This equivalence holds in $\mathbb{R}$ because $\mathbb{R}$ is first-countable: every point has a countable neighborhood base (e.g., $\{B_{1/k}(x_0) : k \in \mathbb{N}\}$). 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.

The key idea is that in first-countable spaces, we can always find a countable collection of neighborhoods that “approximate” any neighborhood, allowing sequences to capture all the limiting behavior.