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

4.1.3 · D2 · Maths › Calculus I — Limits & Derivatives › One-sided limits — left-hand, right-hand

Step 1 — "Approaching" dikhta kaise hai?

KYA. Ek function $f$ lo — ek machine jo $x$ number khaati hai aur $f (x)$ number ugalti hai. Horizontal axis par ek target input $a$ chuno. Hum $x$ ko $a$ ki taraf slide karte hain aur height $f (x)$ dekhte hain.

KYUN. Isse pehle ki hum "approach" ko left aur right mein split karein, humein yeh dekhna hai ki limit height ki journey ke baare mein hoti hai, na ki $a$ par us akele point ke baare mein. "Approaching" ka matlab hai: $a$ ke jitna chahein utna paas jao, bina uspar utare.

TASVEER. Figure mein, chhote dot ko curve ke saath chalao. Jaise uska paon ( $x$ ) $a$ par dashed vertical line ki taraf badhta hai, uski height ( $f (x)$ ) $L$ par dashed horizontal line ki taraf chadti hai.

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

Step 2 — $a$ tak do raaste hain

KYA. Point $x$ , $a$ tak do directions se pahunch sakta hai: chhote numbers se dheere dheere aana (left se), ya bade numbers se neeche aana (right se).

KYUN. Number line par, $a$ apne neeche wale numbers aur upar wale numbers ke beech mein hota hai. " $a$ ko approach karo" asal mein do alag journeys hain. Agar hum kabhi koi cliff, price jump, ya step describe karna chahein, toh humein ek road ke baare mein ek baar mein baat karni chahiye.

TASVEER. Burnt-orange arrow left se aata hai ( $x < a$ : values jaise $a - 0.1, a - 0.01$ ). Teal arrow right se aata hai ( $x > a$ : values jaise $a + 0.1, a + 0.01$ ). Dono $a$ ki taraf aim karte hain par ushe kabhi touch nahi karte.

Step 3 — Jab dono raaste ek hi kamre mein milte hain

KYA. Maano left journey aur right journey dono height ko same value $L$ ki taraf drive karti hain. Tab chahe $x$ kisi bhi tarah $a$ ki taraf chhupta aaye, height $L$ ke paas aa jaati hai.

KYUN. Limit ek vaada hai: " $x$ ko $a$ ke itna paas lao aur $f (x)$ , $L$ ke paas hoga." Agar har raasta yeh vaada maanta hai, toh vaada pura hota hai — ordinary two-sided limit exist karta hai aur $L$ ke barabar hota hai.

TASVEER. Dono arrows same horizontal line $L$ par aakar milte hain. Dekho kaise dono heights $f (x)$ ek saath simat jaati hain jaise $x$ , $a$ par tight hota hai — woh $L$ par haath milaate hain.

Step 4 — Jab dono raaste alag kehte hain (ek jump)

KYA. Ab left road ek height ki taraf jaaye aur right road alag height ki taraf. Koi ek $L$ naam dene wala nahi.

KYUN. Vaada " $f (x)$ $L$ ke paas pahunchta hai" pura nahi ho sakta, kyunki kaun sa $L$ ? Left kehta hai $2$ , right kehta hai $5$ — vaada apne aap se contradict karta hai. Toh two-sided limit fail ho jaata hai, chahe dono one-sided limits apne aap mein bilkul theek hain.

TASVEER. Orange arrow neechi height par aakar milta hai; teal arrow uncchi height par. Unke beech ka gap hi jump hai. Hollow circles mark karte hain ki koi bhi height " $a$ par value" nahi hai — woh point khud skip ho gaya hai.

Step 5 — $a$ par value ek spectator hai

KYA. Sirf $x = a$ par akeli height badlo — welcome mat hatao. Two-sided limit nahi hilegaa.

KYUN. Journey $a$ ke paas hoti hai, aisi $x$ values use karke jo kabhi $a$ ke barabar nahi hoti. Definition literally $x = a$ forbid karti hai (yaad karo $0 < ∣ x - a ∣$ ). Toh $f (a)$ ek spectator hai, player nahi.

TASVEER. Ek smooth curve $L$ ki taraf jaati hai, lekin $a$ par point ko khींchkar height $7$ par akela dot bana diya gaya hai. Dono sides ke arrows abhi bhi $L$ par milte hain — woh akela dot kuch nahi badalta.

Step 6 — $ε$ – $δ$ machine jo rule prove karta hai

KYA. Ab hum "L ke paas pahunchta hai" ko exact banate hain. Tolerance $ε$ (height mein kitni closeness chahiye) ek window $δ$ force karta hai (input mein kitni closeness chahiye).

KYUN. "Paas" jaise words phisalne wale hain. ε–δ definition unhe ek challenge–response game se replace karta hai, aur yeh reveal karta hai ki kyun dono sides ko agree karna chahiye: two-sided rule literally do one-sided rules hain jo AND word se jude hain.

TASVEER. $L$ ke around half-height $ε$ ka ek horizontal band, aur $a$ ke around half-width $δ$ ka ek vertical band. Two-sided band ( $0 < ∣ x - a ∣ < δ$ ) left half aur right half mein split hota hai. Sirf ek half rakhna ek one-sided limit hai.

Step 7 — Degenerate cases jinmein kabhi mat phansna

KYA. Teen special situations jahan "ek side" oddly behave karti hai. Har ek dikhayi jaati hai taaki koi reader kisi unseen scenario mein na phas jaaye.

KYUN. Real functions mein edges, poles, aur steps hote hain. Rule abhi bhi unhe govern karta hai — lekin "exists" ke saath dhyan chahiye.

TASVEER. Teen panels: (a) ek domain edge ( $x$ at $0$ : $0$ ki left mein chalne ki jagah nahi); (b) ek pole ( $1/ x$ at $0$ : right $+ \infty$ tak jaata hai, left $- \infty$ tak girta hai); (c) floor $⌊ x ⌋$ at $3$ (right $3$ par hai, left $2$ par).

Ek-tasveer summary

Yeh single frame poori kahani stack karta hai: do raaste (orange left, teal right), shared- $L$ handshake (upar), disagreeing jump (beech mein), aur spectator point $f (a)$ jo kabhi participate nahi karta.

Recall Feynman retelling — poora walkthrough seedhi baaton mein

Socho position $a$ par ek darwaaza hai. Tum ustak left hallway (numbers $a$ se chhote) ya right hallway (numbers $a$ se bade) se chal ke aa sakte ho. Ek one-sided limit poochta hai: "sirf is ek hallway se aate hue, darwaaze ke theek pehle main kis kamre ki taraf ja raha hoon?" Agar dono hallways tumhe same room $L$ dikhayein, woh kamra hi limit hai — dono hallways haath milaate hain (Steps 1–3). Agar left kitchen dikhaaye aur right bathroom, toh "darwaaze ke peeche ek room" nahi hai, toh two-sided limit fail hota hai — chahe har hallway ka view bilkul clear tha (Step 4). Jo cheez exactly darwaaze par hoti hai — welcome mat $f (a)$ — kabhi nahi badlaati ki tum kis kamre ki taraf chal rahe the (Step 5). ε–δ game "ki taraf ja raha" ko exact banata hai: demand karo ki height $L$ ke $ε$ ke andar rahe, aur tum hamesha ek inputs ki window $δ$ dhoondh sakte ho jo apna vaada rakhti hai — aur two-sided window sirf left window AND right window hai jo judi hain (Step 6). Aakhir mein, kuch darwaaze odd hain: sirf ek hallway wali diwar ( $x$ ), ek hallway jo cliff se $\pm \infty$ par gir jaati hai ( $1/ x$ ), ya ek staircase jo har step par jump karta hai ( $⌊ x ⌋$ ) — Step 7. Same rule sabko govern karta hai: dono roads exist karni chahiye aur match karni chahiye.

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