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

$$ f(x) = x^2 $$

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

Let $\{a_n\}_{n=m}^{\infty}$ be a sequence of reals. Think of $m$ as the starting index of the sequence. Here are some sequences with different starting indices:

For $m=1$:

$$ a_1, a_2, a_3, a_4, a_5, \ldots $$

For $m=3$:

$$ a_3, a_4, a_5, a_6, a_7, a_8, \ldots $$

Now consider a natural number $N \geq m$. A tail of our sequence $\{a_n\}_{n=m}^{\infty}$ is defined as:

$$\{a_n\}_{n=N}^{\infty}$$

Visual Illustration

Let’s take $m=1$ and consider the original sequence starting from index 1:

If you interrogate the formula for the dot product, it will gift to you all the explanations you need. You’ve seen mathematicians define the same dot product in different ways.

The Geometric Definition of the Dot Product

In the two-dimensional plane, start with two vectors at the origin. Think of our vectors as line segments that we draw starting at coordinate $(0,0)$ (your origin). The dot product of those two line segments (vectors) equals:

THROW YOUR RULER, an integral will be all you need to measure distance. In fact, the integral can measure distances your ruler cannot.

You’re an engineer tasked with measuring the distance traveled by a rollercoaster on a parabolic path. Let the function $y = x^2$ on the interval $[-1, 1]$ represent that parabola. Keep in mind that the rollercoaster doesn’t exist yet, so we cannot directly go and measure it. We only know that it will follow the shape of the above quadratic.