Foundations — One-sided limits — left-hand, right-hand

This page assumes nothing. Before you can read the parent note One-sided Limits, you must be able to read every squiggle it uses without flinching. So we build each symbol from the ground up: plain words first, then a picture, then why the topic cannot live without it.

0. The number line — the stage everything happens on

Look at Figure 1 below. The horizontal ruler runs left-to-right; the red dot marks the target $a=3$; the blue arrow reminds you that "smaller / left" is one way and the orange arrow that "larger / right" is the other. Everything on this whole page happens on this one line.

WHY the topic needs this: the whole subject is about motion along this line — sliding $x$ toward the target $a$. "Left" and "right" (the heart of one-sided limits) are literally directions on this ruler. If you cannot see the line, "from the left" is just noise.

1. A function $f(x)$ — the height machine

In Figure 2, the red dot on the horizontal axis is the input $x$; follow the dashed line straight up to the blue curve, then across to the vertical axis — that orange dot is the output height $f(x)$. Reading a graph is always this "up then across" move.

• $x$ ::: the input, a position on the horizontal line.
• $f(x)$ ::: the output height the machine gives back for that input.

WHY the topic needs this: a limit asks "where is the height $f(x)$ heading as the input $x$ moves?" No height machine, no thing to head toward.

2. The target $a$ and the arrow $\to$ ("approaches")

WHY the topic needs this: one-sided limits are entirely about how $x$ approaches $a$ — the direction of that slide. The arrow is the verb of the whole sentence.

3. The concept: what a one-sided limit is

Before we meet its notation, let's name the idea in plain words.

The ordinary (two-sided) limit lets $x$ come from either side. A one-sided limit deliberately blocks one direction and watches what happens using only the other. That single restriction is the whole new idea of this topic.

4. The little $+$ and $-$ superscripts — direction of approach

Now that we know what a one-sided limit is, here is the notation that trips everyone up first.

Figure 3 shows both approaches on one ruler. The blue dots creep in toward $a$ from the left (these are the $a^{-}$ inputs, all smaller than $a$); the orange dots creep in from the right (the $a^{+}$ inputs, all larger than $a$). Watch the arrows point inward toward the red target from each side.

WHY the topic needs this: these two superscripts are how we write down the one-sided idea from Section 3. They split "approach" into two labelled directions.

5. The word "limit" and the symbol $\lim$

The one-sided versions just swap in the direction superscript:

WHY the topic needs this: $\lim$ is the noun of the whole chapter. Everything else — continuity, derivatives, asymptotes — is built out of this word. See Limit of a function — intuitive & ε-δ definition for the deeper machinery.

6. The symbol $L$ — the value approached

• $L$ ::: the single height value the outputs $f(x)$ pile up around as $x\to a$.

WHY the topic needs this: the punchline of the topic is "the two-sided limit exists iff both sides aim at the same $L$." Without a name for "the value approached," you cannot state that rule.

7. The word "iff" and the symbol $⟺$

The parent's boxed rule uses it: ${\lim }_{x\to a}f(x)=L⟺{\lim }_{x\to a^{-}}f(x)={\lim }_{x\to a^{+}}f(x)=L$

This says: "the two-sided limit equals $L$" is exactly the same statement as "both one-sided limits exist and both equal $L$." Not one direction — a two-way street.

WHY the topic needs this: it packages the entire test for a two-sided limit into one honest logical equivalence. It is the topic's headline theorem.

8. Absolute value $|x|$ and $\infty$ — two guests in the examples

The parent's worked examples quietly use two more symbols. Let's earn them.

Figure 4 puts the $|x|$ idea to work: it graphs the sign function $f(x)=|x|/x$. Notice the blue level sitting at $-1$ for every $x$ to the left of zero, and the orange level at $+1$ for every $x$ to the right. The two hollow dots at $x=0$ show the function never actually lands there — and because the two sides sit at different heights, this is our cleanest picture of "left ≠ right."

WHY the topic needs these: $|x|$ builds the cleanest example of a left≠right mismatch, and $\infty$ names what happens at a pole. Both appear in the parent's examples, so both must be zero-to-hero clear.

9. Piecewise definition — the big curly brace $\{$

WHY the topic needs this: the cleanest place to see two sides disagree is where the rule itself switches. Piecewise notation is how we write such functions down.

Prerequisite map

Here is how all nine foundations feed into the topic. Read the diagram top-down: the boxes near the top are the raw ingredients (number line, height machine), the middle boxes combine them (approach, direction superscripts, the word limit), and the bottom boxes are the payoff (one-sided limits, the two-sided rule, the worked examples). Each arrow means "you need the box behind the arrow before the box ahead of it makes sense."

Equipment checklist

Test yourself — cover the right side and answer before revealing.

Connections

• Limit of a function — intuitive & ε-δ definition — the parent machinery of $\lim$.
• Continuity at a point — needs ${\lim }_{x\to a}f=f(a)$, built from these symbols.
• Jump, removable & infinite discontinuities — what mismatched sides cause.
• Vertical asymptotes — where $\infty$ shows up.
• Greatest integer / floor function — a piecewise staircase example.
• Differentiability — left & right derivatives — same left/right idea applied to slopes.