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

What exactly is a one-sided limit?

WHAT the symbols mean: $a^{-}$ is not a number you plug in — it is shorthand for " $x$ approaching $a$ while staying less than $a$ ". You never reach $a$ itself; the value $f(a)$ is irrelevant to the limit.

Deriving the rule from first principles

HOW to see it with the $\varepsilon$ – $\delta$ definition. Two-sided says: $\forall \varepsilon>0,\ \exists \delta>0 : 0<|x-a|<\delta \implies |f(x)-L|<\varepsilon.$ The condition $0<|x-a|<\delta$ splits into two disjoint pieces: $\underbrace{-\delta<x-a<0}_{\text{left: } x<a}\quad\text{or}\quad\underbrace{0<x-a<\delta}_{\text{right: } x>a}.$ The one-sided definitions keep only one piece: $\lim_{x\to a^+}f(x)=L \iff \forall\varepsilon\,\exists\delta:\ 0<x-a<\delta \implies |f(x)-L|<\varepsilon,$ $\lim_{x\to a^-}f(x)=L \iff \forall\varepsilon\,\exists\delta:\ -\delta<x-a<0 \implies |f(x)-L|<\varepsilon.$ Because the two-sided condition is exactly the logical AND of the two halves, the two-sided limit holds iff both halves hold with the same $L$ . That is the boxed rule — derived, not memorised.

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

Worked examples

Forecast-then-Verify drill

Recall Predict first, then check

Before reading: for $f(x)=\sqrt{x}$ at $x=0$ , which one-sided limit even makes sense? Answer: Only the right limit exists, $\lim_{x\to0^+}\sqrt{x}=0$ , because $\sqrt{x}$ is undefined for $x<0$ (real domain). The two-sided limit does not exist because the left side has no domain. Edge of domain ⇒ only one side is available.

Common mistakes (Steel-manned)

80/20 — the few things that carry the rest

Recall Feynman: explain to a 12-year-old

Imagine walking toward a doorway. You can come at it from the left hallway or from the right hallway. A one-sided limit asks "what do you see right before you reach the door, coming from this one side?" If both hallways show you the same room through the door, that's the real limit. If the left hallway shows a kitchen and the right shows a bathroom, there's no single "the room behind the door" — the limit doesn't exist. And what's exactly at the doorstep (a welcome mat, $f(a)$ ) doesn't change what room you were heading toward.

Flashcards

What does

\lim_{x\to a^+}f(x)

mean?

The value

f(x)

approaches as

x\to a

using only

x>a

(from the right/above).

When does

\lim_{x\to a}f(x)

exist?

Iff both one-sided limits exist AND are equal:

\lim_{x\to a^-}=\lim_{x\to a^+}

Does

x\to5^-

involve negative

x

No — it means

x

approaches

5

through values less than

5

(e.g.

4.99

), which are positive.

For

f(x)=|x|/x

, the one-sided limits at

0

are?

Right

=+1

, Left

=-1

; two-sided limit does not exist.

Does

f(a)

affect the limit at

a

No, the limit depends only on the neighbourhood of

a

, not the value

f(a)

Why does

\lim_{x\to0}\sqrt{x}

not exist (two-sided)?

\sqrt{x}

is undefined for

x<0

, so the left-hand limit has no domain; only

\lim_{x\to0^+}=0

exists.

\lim_{x\to3^-}\lfloor x\rfloor

and

\lim_{x\to3^+}\lfloor x\rfloor

2

and

3

respectively; they disagree so no two-sided limit at integers.

\lim_{x\to0^+}1/x=+\infty

an existing limit?

Strictly no —

+\infty

isn't a real number; the limit diverges to

+\infty

Connections

Limit of a function — intuitive & ε-δ definition
Continuity at a point (needs $\lim_{x\to a}f=f(a)$ , so one-sided limits matter)
Jump, removable & infinite discontinuities
Vertical asymptotes (one-sided limits to $\pm\infty$ )
Greatest integer / floor function
Differentiability — left & right derivatives (same one-sided idea on slopes)

Concept Map

Hinglish (regional understanding)

Intuition Hinglish mein samjho

Dekho, ek normal limit poochta hai: "jaise $x$ point $a$ ke paas jaata hai, $f(x)$ kahan ja raha hai?" Lekin $x$ do tareeke se $a$ ke paas aa sakta hai — left se (chote values, $x<a$ ) ya right se (bade values, $x>a$ ). One-sided limit sirf ek hi direction ke liye answer deta hai. Right-hand limit ko hum $\lim_{x\to a^+}f(x)$ likhte hain (plus matlab upar/right se), aur left-hand ko $\lim_{x\to a^-}f(x)$ (minus matlab neeche/left se).

Sabse important rule: poora (two-sided) limit tabhi exist karta hai jab dono one-sided limits exist karein aur equal ho. Soch lo do galiyaan ek darwaaze pe milti hain — agar dono galiyon se same kamra dikhta hai, tabhi "darwaaze ke peeche ka kamra" defined hai. Agar left se kitchen aur right se bathroom dikhe (jaise $|x|/x$ mein $-1$ aur $+1$ ), toh single limit possible hi nahi.

Ek aur trap: $a^-$ ka matlab negative number nahi hai! $x\to5^-$ ka matlab hai $x=4.9, 4.99$ — yeh sab positive hain, bas $5$ se chote hain. Aur yaad rakho — limit ke liye $f(a)$ ki value matter nahi karti, sirf aas-paas ka behaviour matter karta hai (Example 2 mein $g(2)=7$ tha par limit $3$ thi).

Yeh concept aage bahut kaam aata hai — continuity, jumps, vertical asymptotes ( $1/x$ jaise jahan ek side $+\infty$ aur doosri $-\infty$ jaati hai), aur domain ke edge pe (jaise $\sqrt{x}$ pe $0$ ke left mein function exist hi nahi karta, toh sirf right limit milti hai). Master kar lo, toh aadha calculus easy ho jaata hai.

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

What exactly is a one-sided limit?

Deriving the rule from first principles

Worked examples

Forecast-then-Verify drill

Common mistakes (Steel-manned)

80/20 — the few things that carry the rest

Flashcards

Connections

Concept Map

Hinglish (regional understanding)

Go deeper — visual, from zero

Test yourself — Calculus I — Limits & Derivatives

Connections