Consider the function $f: \mathbb{N} \to \mathbb{R}$ where

This is a simple quadratic equation you’re acquainted with, but note that our domain are only natural numbers, meaning $1, 2, 3, 4, \ldots$. The graph of this function looks like this:

But this may not be the graph of the most common version of this quadratic. Indeed, it’s more commonly defined as a function from $f: \mathbb{R} \to \mathbb{R}$. While the natural numbers are said to be countable, the real numbers are uncountable.

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.

The outcomes of a sample space (see the article on sample spaces referenced earlier) are not necessarily numerical. However, as mathematicians, we are interested in quantities such as averages, variances, and distributions, all of which require numbers.

This is where random variables come in.

A random variable is a function that maps outcomes of a sample space to real numbers. If the sample space is denoted by $\Omega$, then a random variable is a mapping