Visual walkthrough — One-sided limits — left-hand, right-hand

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Step 1 — What does "approaching" even look like?

WHAT. Pick a function $f$ — a machine that eats a number $x$ and spits out a number $f(x)$. Pick a target input $a$ on the horizontal axis. We slide $x$ toward $a$ and watch the height $f(x)$.

WHY. Before we can split "approach" into left and right, we need to see that a limit is about the journey of the height, never about the single point at $a$. The word "approaching" means: get as close as you like to $a$ without landing on it.

PICTURE. In the figure, walk the little dot along the curve. As its foot ($x$) marches toward the dashed vertical line at $a$, its height ($f(x)$) climbs toward the dashed horizontal line at $L$.

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Step 2 — There are two roads to $a$

WHAT. The point $x$ can reach $a$ from two directions: creeping up from smaller numbers (the left) or coming down from larger numbers (the right).

WHY. On a number line, $a$ sits between the numbers below it and the numbers above it. "Approach $a$" is secretly two separate journeys. If we ever want to describe a cliff, a price jump, or a step, we must be allowed to talk about one road at a time.

PICTURE. The burnt-orange arrow comes from the left ($x<a$: values like $a-0.1,a-0.01$). The teal arrow comes from the right ($x>a$: values like $a+0.1,a+0.01$). Both aim at $a$ but never touch it.

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Step 3 — When both roads meet the same room

WHAT. Suppose the left journey and the right journey both drive the height toward the same value $L$. Then no matter how $x$ sneaks toward $a$, the height ends up near $L$.

WHY. A limit is a promise: "get $x$ close enough to $a$ and $f(x)$ will be near $L$." If every road obeys the promise, the promise is kept — the ordinary two-sided limit exists and equals $L$.

PICTURE. Both arrows land on the same horizontal line $L$. Watch the two heights $f(x)$ squeeze together as $x$ tightens on $a$ — they shake hands at $L$.

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Step 4 — When the two roads disagree (a jump)

WHAT. Now let the left road head to one height and the right road head to a different height. There is no single $L$ to name.

WHY. The promise "$f(x)$ ends up near $L$" cannot be kept, because which $L$? Left says $2$, right says $5$ — the promise contradicts itself. So the two-sided limit fails, even though both one-sided limits are perfectly fine on their own.

PICTURE. The orange arrow lands on the lower height; the teal arrow lands on the higher height. The gap between them is the jump. The hollow circles mark that neither height is "the value at $a$" — the point itself is skipped.

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Step 5 — The value at $a$ is a spectator

WHAT. Change only the single height at $x=a$ — move the welcome mat. The two-sided limit does not budge.

WHY. The journey happens near $a$, using $x$ that never equal $a$. The definition literally forbids $x=a$ (recall $0<|x-a|$). So $f(a)$ is a spectator, not a player.

PICTURE. A smooth curve heading to $L$, but the point at $a$ has been yanked up to a lone dot at height $7$. The arrows from both sides still meet at $L$ — the stray dot changes nothing.

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Step 6 — The $\epsilon$–$\delta$ machine that proves the rule

WHAT. We now make "ends up near $L$" exact. The tolerance $\epsilon$ (how close in height we demand) forces a window $\delta$ (how close in input we need).

WHY. Words like "near" are slippery. The ε–δ definition replaces them with a challenge–response game, and it reveals why both sides must agree: the two-sided rule is literally the two one-sided rules glued with the word AND.

PICTURE. A horizontal band of half-height $\epsilon$ around $L$, and a vertical band of half-width $\delta$ around $a$. The two-sided band ($0<|x-a|<\delta$) splits into a left half and a right half. Keeping only one half is a one-sided limit.

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Step 7 — The degenerate cases you must never trip over

WHAT. Three special situations where "one side" behaves oddly. Each gets shown so no reader hits an unshown scenario.

WHY. Real functions have edges, poles, and steps. The rule still governs them — but "exists" needs care.

PICTURE. Three panels: (a) a domain edge ($\sqrt{x}$ at $0$: nothing left of $0$ to walk on); (b) a pole ($1/x$ at $0$: right shoots to $+\infty$, left plunges to $-\infty$); (c) the floor $\lfloor x\rfloor$ at $3$ (right sits at $3$, left sits at $2$).

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The one-picture summary

This single frame stacks the whole story: the two roads (orange left, teal right), the shared-$L$ handshake (top), the disagreeing jump (middle), and the spectator point $f(a)$ that never joins in.

Recall Feynman retelling — the whole walkthrough in plain words

Picture a doorway at position $a$. You can walk up to it down the left hallway (numbers smaller than $a$) or the right hallway (numbers bigger than $a$). A one-sided limit asks: "coming down this one hallway, what room am I heading toward just before the door?" If both hallways show you the same room $L$, that room is the limit — the two hallways shake hands (Steps 1–3). If the left shows a kitchen and the right a bathroom, there is no single "room behind the door", so the two-sided limit fails — even though each hallway had a perfectly clear view (Step 4). Whatever sits exactly on the doorstep — a welcome mat $f(a)$ — never changes what room you were walking toward (Step 5). The ε–δ game makes "heading toward" exact: demand the height stay within $\epsilon$ of $L$, and you can always find a window $\delta$ of inputs that keeps its promise — and the two-sided window is just the left window AND the right window glued together (Step 6). Finally, some doorways are odd: a wall with only one hallway ($\sqrt{x}$), a hallway that drops off a cliff to $\pm \infty$ ($1/x$), or a staircase that jumps at every step ($\lfloor x\rfloor$) — Step 7. Same rule governs them all: both roads must exist and match.

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Connections

• One-sided limits — left-hand, right-hand (parent)
• Limit of a function — intuitive & ε-δ definition
• Continuity at a point
• Jump, removable & infinite discontinuities
• Vertical asymptotes
• Greatest integer / floor function
• Differentiability — left & right derivatives