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

4.1.3 · Maths › Calculus I — Limits & Derivatives

Exactly kya hota hai ek one-sided limit?

SYMBOLS ka matlab: $a^{-}$ koi aisa number nahi hai jisko tum plug in karo — yeh shorthand hai " $x$ , $a$ ko approach kar raha hai jabki $a$ se chhota rehta hai". Tum kabhi $a$ tak pahunchte nahi; value $f (a)$ limit ke liye irrelevant hai.

Rule ko first principles se derive karna

$ε$ – $δ$ definition se kaise dekho. Two-sided kehti hai: $\forall ε > 0, \exists δ > 0 : 0 < ∣ x - a ∣ < δ ⟹ ∣ f (x) - L ∣ < ε .$ Condition $0 < ∣ x - a ∣ < δ$ do alag pieces mein split hoti hai: $left: x < a - δ < x - a < 0 or right: x > a 0 < x - a < δ .$ One-sided definitions sirf ek piece rakhti hain: $lim_{x \to a^{+}} f (x) = L ⟺ \forall ε \exists δ : 0 < x - a < δ ⟹ ∣ f (x) - L ∣ < ε,$ $lim_{x \to a^{-}} f (x) = L ⟺ \forall ε \exists δ : - δ < x - a < 0 ⟹ ∣ f (x) - L ∣ < ε .$ Kyunki two-sided condition bilkul dono halves ka logical AND hai, two-sided limit tab hold hoti hai jabhi dono halves same $L$ ke saath hold karein. Yahi boxed rule hai — derived hai, memorise nahi kiya.

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

Worked examples

Forecast-then-Verify drill

Recall Pehle predict karo, phir check karo

Padhne se pehle: $f (x) = x$ ke liye $x = 0$ par, kaun sa one-sided limit make sense karta hai? Jawab: Sirf right limit exist karta hai, $lim_{x \to 0^{+}} x = 0$ , kyunki $x$ undefined hai $x < 0$ ke liye (real domain). Two-sided limit exist nahi karta kyunki left side ka koi domain nahi hai. Domain ki edge ⇒ sirf ek side available hai.

Common mistakes (Steel-manned)

80/20 — woh thodi si cheezein jo baaki sab carry karti hain

Recall Feynman: ek 12-saal ke bachche ko samjhao

Socho tum ek doorway ki taraf chal rahe ho. Tum uske paas left hallway se aa sakte ho ya right hallway se. Ek one-sided limit poochta hai "door tak pahunchne se theek pehle tum kya dekhte ho, is ek side se aate hue?" Agar dono hallways tumhe same room dikhate hain door ke through, toh wahi real limit hai. Agar left hallway ek kitchen dikhata hai aur right ek bathroom, toh koi single "the room behind the door" nahi hai — limit exist nahi karti. Aur jo cheez exactly doorstep par hai (ek welcome mat, $f (a)$ ) yeh nahi badlta ki tum kis room ki taraf ja rahe the.

Flashcards

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

ka kya matlab hai?

Woh value jisko

f (x)

approach karta hai jab

x \to a

sirf

x > a

use karke (right/above se).

lim_{x \to a} f (x)

kab exist karta hai?

Tabhi jab dono one-sided limits exist karein AUR equal hon:

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

Kya

x \to 5^{-}

mein negative

x

involved hai?

Nahi — matlab hai

x

5

5

se chhoti values ke through approach karta hai (jaise

4.99

), jo positive hain.

f (x) = ∣ x ∣/ x

ke liye

0

par one-sided limits kya hain?

Right

= + 1

, Left

= - 1

; two-sided limit exist nahi karta.

Kya

f (a)

a

par limit ko affect karta hai?

Nahi, limit sirf

a

ki neighbourhood par depend karti hai, value

f (a)

par nahi.

lim_{x \to 0} x

(two-sided) exist kyun nahi karta?

x

undefined hai

x < 0

ke liye, toh left-hand limit ka koi domain nahi hai; sirf

lim_{x \to 0^{+}} = 0

exist karta hai.

lim_{x \to 3^{-}} ⌊ x ⌋

aur

lim_{x \to 3^{+}} ⌊ x ⌋

Krama se

2

aur

3

; yeh disagree karte hain toh integers par koi two-sided limit nahi.

Kya

lim_{x \to 0^{+}} 1/ x = + \infty

ek existing limit hai?

Strictly nahi —

+ \infty

koi real number nahi hai; limit

+ \infty

ki taraf diverge karti hai.

Connections

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

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