Foundations — Limit laws — sum, product, quotient, constant multiple

Before you can trust the limit laws, you must be fluent in every mark on the page. This note takes each symbol the parent uses — one at a time, from nothing — pairs it with a picture, and says why the topic cannot live without it. Read top to bottom: each item leans on the one above it.

1. The number line and a point $a$

The symbol $a$ (or $2$, or $3$) is the target we are walking toward. It is a fixed, unmoving dot. The topic needs it because every limit question begins "as $x$ approaches..." — approaches what? — the point $a$.

2. The four arithmetic operations $+,-,\times ,÷$

Because the whole topic is about limits "flowing through" arithmetic, we must first be crisp about what those four operations are and what each looks like on the number line.

Why the topic needs it: the limit laws are literally named Sum, Constant multiple, Product, Quotient — one law per operation. To say "the limit passes through $\times$", you must already know that $\times$ means "scale/repeat". Division carries the famous fine print ($M\ne 0$) precisely because you cannot share into zero equal parts.

3. The variable $x$ and "approaches" ($\to$)

Why "approaches" and not "equals"? Because a limit asks about the journey, not the destination. The whole power of Example 3 in the parent — cancelling $\frac{x-3}{x-3}$ near $x=3$ — only works because $x\ne 3$ along the way. If the arrow meant "equals", that cancellation would be illegal. The gap between "$x$ near $a$" and "$x=a$" is the entire reason limits exist.

4. A function $f$ and its output $f(x)$

Picture a graph: the horizontal axis is the input road, the vertical axis is the output height. Above each input $x$ on the floor sits a single point at height $f(x)$; joined up, these points draw the curve.

Why the topic needs it: the limit laws are all about combining outputs. When we write $f+g$, we mean "run both machines on the same $x$ and add their outputs" — using the addition of Section 2. You cannot combine outputs before you can name them, so $f(x)$ and $g(x)$ come first.

5. The limit symbol $\underset{x\to a}{\lim }f(x)=L$

Look at the figure: as the input walker slides toward $a$ (from both sides), the point on the curve slides toward one height $L$. That height is the limit.

Note the open circle on the curve at $x=a$ in the figure — it marks that we don't care what happens at $a$ itself, only what the curve is aiming at nearby.

Why we need a name $L$: the laws are statements about $L$ and $M$. To say "the sum heads to $L+M$", we must first have the symbols $L$ and $M$ standing for "where each piece is heading".

6. Absolute value $|\cdot |$ — the "distance" symbol

Why the topic needs it: "close to" must become a number we can shrink. In the parent's proofs, $|f(x)-L|$ means "how far the output is from its target $L$", and $|x-a|$ means "how far the input is from $a$". The whole game is: keep $|x-a|$ small, and $|f(x)-L|$ stays small. Without a distance symbol you cannot say "small".

This is the single tool the parent's Sum-law proof leans on — it lets us bound the combined error $|(f-L)+(g-M)|$ by the two separate errors $|f-L|+|g-M|$, each of which we already know how to control.

7. $\epsilon$ and $\delta$ — the twin tolerances

The full sentence behind ${\lim }_{x\to a}f(x)=L$ is:

$0<|x-a|<\delta \Rightarrow |f(x)-L|<\epsilon .$

Read piece by piece with the symbols you now own:

• $|x-a|<\delta$: the input is within distance $\delta$ of $a$.
• $0<|x-a|$: but not equal to $a$ (that's the "approaches, not equals" rule again).
• $\Rightarrow$: "forces" / "guarantees".
• $|f(x)-L|<\epsilon$: then the output is within distance $\epsilon$ of $L$.

Why the topic needs this: it turns the vague word "heading toward" into a challenge game we can actually prove things about. This precise definition is what the parent's Sum, Product, and Quotient proofs manipulate. Full detail lives in Epsilon-Delta definition of a limit.

8. Constants $c$, $n$, and grouping symbols

Before the atomic limits, one last batch of plain bookkeeping marks:

These are bookkeeping marks, but the reader must not stumble on them mid-proof — so they are named here once and for all.

9. The two atomic limits

Everything the parent builds rests on two facts so simple they need no laws:

Why they matter: the laws only combine limits you already know. These two are the "already known" seeds. Add and multiply them with the laws and you can reach every polynomial and rational limit in the parent.

Prerequisite map

The diagram below (in Mermaid, a plain text-to-flowchart syntax — each --> is an arrow "feeds into") shows how the foundations stack up. In words: the number line gives us the point $a$ and the four operations; from $a$ we build "$x$ approaches $a$", then functions and their outputs, then the limit $L$. Separately, absolute value and the triangle inequality feed the $\epsilon$–$\delta$ game. The limit idea plus the $\epsilon$–$\delta$ game give the two atomic limits, and everything together powers the limit laws.

Equipment checklist

Each line below has the form Question ::: Answer. The ::: is a reveal separator — cover everything after it, try to answer, then check yourself against the hidden text.

Connections

• Epsilon-Delta definition of a limit — where the $\epsilon$–$\delta$ game of Section 7 is developed in full.
• One-sided limits — the "from the left / from the right" walk of Section 3, split apart.
• Continuity — the case where $L$ does equal $f(a)$.
• Parent: Limit laws — sum, product, quotient, constant multiple.