Calculus for AI Beginners, Part 5: Definition and Properties of Limits

Limits describe trends as inputs approach a point, not just the function’s value at that point. Understanding limits is foundational for grasping continuity and derivatives.

We examine how the function behaves as we approach the point from the left and from the right separately. If the left-hand and right-hand trends differ, the (two-sided) limit does not exist.

In the previous article, we introduced the basic concepts and notation of functions and limits. This article delves deeper into the definition and properties of limits—a cornerstone for understanding subsequent topics such as continuity and differentiability.

Definition of a Limit

The limit is a core concept in calculus, used to describe how a function behaves near a particular point. Specifically, saying “the limit of $f(x)$ as $x$ approaches $c$ is $L$” means that as $x$ gets arbitrarily close to $c$, the output $f(x)$ gets arbitrarily close to $L$. Formally, we write:

When learning the definition of a limit, first develop intuition: observe whether function values stabilize as inputs gradually approach a target point. This intuition underpins later concepts like continuity and derivatives.

$$\lim_{x \to c} f(x) = L$$

This means that no matter how $x$ approaches $c$ (e.g., from left, right, or oscillating), $f(x)$ must get arbitrarily close to $L$.

The $\epsilon$–$\delta$ Definition

A rigorous, formal definition of limits uses the symbols $\epsilon$ (epsilon) and $\delta$ (delta). Precisely, we say

$$\lim_{x \to c} f(x) = L$$

if and only if:
For every $\epsilon > 0$, there exists a $\delta > 0$ such that
whenever $0 < |x - c| < \delta$, it follows that $|f(x) - L| < \epsilon$.

This definition ensures mathematical precision and eliminates ambiguity about what “approaching” truly means.

Case Study

Consider $f(x) = 2x$. We want to compute its limit as $x \to 3$:

$$\lim_{x \to 3} f(x)$$

By direct substitution, as $x$ approaches 3, $f(x) = 2x$ approaches 6. We verify this using the $\epsilon$–$\delta$ definition:

1. Choose $\epsilon = 0.1$. We need to find $\delta > 0$ such that $|f(x) - 6| < 0.1$.
2. Since $f(x) = 2x$, $$|2x - 6| < 0.1 \quad \Rightarrow \quad |x - 3| < 0.05.$$
3. So we may take $\delta = 0.05$.

This shows: whenever $x \in (2.95,\, 3.05)$, then $f(x) \in (5.9,\, 6.1)$ — satisfying the limit condition exactly.

Properties of Limits

Limits obey several important algebraic properties—essential tools for computing derivatives and integrals. Here are key ones:

While reading AI-Ready Calculus for Beginners: Functions & Limits — Defining and Understanding Limits, first identify the practical scenario you aim to solve, then connect core concepts with concrete practice steps. This prevents getting lost in isolated terminology when diving into details.

1. Linearity of Limits

If $\displaystyle \lim_{x \to c} f(x) = L$ and $\displaystyle \lim_{x \to c} g(x) = M$, then for any constants $a$ and $b$:

• Linear combination:

$$\lim_{x \to c} \big( a f(x) + b g(x) \big) = aL + bM$$

2. Product Rule for Limits

If both $f(x)$ and $g(x)$ have limits at $c$, then their product does too:

$$\lim_{x \to c} \big( f(x)\,g(x) \big) = L \cdot M$$

3. Quotient Rule for Limits

If $\displaystyle \lim_{x \to c} g(x) \neq 0$, and both $f(x)$ and $g(x)$ have limits at $c$, then:

$$\lim_{x \to c} \frac{f(x)}{g(x)} = \frac{L}{M}$$

Application Example

Evaluate:

$$\lim_{x \to 2} \frac{x^2 - 4}{x - 2}$$

Direct substitution yields $0/0$ — undefined. Instead, factor and simplify:

1. Factor numerator:

   $$\frac{x^2 - 4}{x - 2} = \frac{(x - 2)(x + 2)}{x - 2}, \quad \text{for } x \ne 2$$

2. Cancel common factor:

   $$x + 2$$

3. Therefore,

   $$\lim_{x \to 2} (x + 2) = 4$$

Thus, the limit exists and equals 4.

When reviewing AI-Ready Calculus for Beginners: Functions & Limits — Defining and Understanding Limits, place key concepts, procedural steps, and observable outcomes on the same page for efficient recall.

When practicing AI-Ready Calculus for Beginners: Functions & Limits — Defining and Understanding Limits, write input conditions, computational actions, and resulting outputs together—making future review faster and more reliable.

Summary

In this article, we explored the formal definition of limits, their fundamental algebraic properties, and illustrative examples. These ideas form the bedrock for understanding continuity and differentiability. In the next article, we’ll build on this foundation to examine how limits enable precise definitions of continuous functions and derivatives. We hope this material strengthens your intuitive and technical grasp of calculus.