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

4.1.3 · D1 · Maths › Calculus I — Limits & Derivatives › One-sided limits — left-hand, right-hand

Is page par kuch bhi assume nahi kiya gaya. Parent note One-sided Limits padhne se pehle, tumhe usme use hone wala har squiggle bina jhijhke padhna aana chahiye. Isliye hum har symbol ground up se build karte hain: pehle plain words, phir ek picture, phir yeh topic uske bina kyu nahi chal sakta.

0. Number line — wo stage jahan sab kuch hota hai

Neeche Figure 1 dekho. Horizontal ruler left-to-right chalti hai; red dot target $a=3$ mark karta hai; blue arrow yaad dilata hai ki "smaller / left" ek direction hai aur orange arrow ki "larger / right" doosra direction hai. Is poore page par sab kuch isi ek line par hota hai.

YEH TOPIC ISKO KYU CHAHIYE: poora subject is line par motion ke baare mein hai — $x$ ko target $a$ ki taraf slide karna. "Left" aur "right" (one-sided limits ki jaan) literally is ruler par directions hain. Agar tum line nahi dekh sakte, to "from the left" sirf shor hai.

1. Ek function $f(x)$ — height machine

Figure 2 mein, horizontal axis par red dot input $x$ hai; dashed line seedha upar blue curve tak jaao, phir vertical axis ki taraf — woh orange dot output height $f(x)$ hai. Graph padhna hamesha yahi "upar phir across" wali move hai.

• $x$ ::: input, horizontal line par ek position.
• $f(x)$ ::: output height jo machine us input ke liye return karti hai.

YEH TOPIC ISKO KYU CHAHIYE: ek limit poochtha hai "jaise input $x$ move karta hai, height $f(x)$ kahan ja rahi hai?" Height machine nahi, to kuch bhi head karne wala nahi.

2. Target $a$ aur arrow $\to$ ("approaches")

YEH TOPIC ISKO KYU CHAHIYE: one-sided limits poori tarah is baare mein hain ki $x$ $a$ ki taraf kaise approach karta hai — us slide ki direction. Arrow poore sentence ki verb hai.

3. Concept: ek one-sided limit kya hota hai

Notation se milne se pehle, idea ko plain words mein naam dete hain.

Ordinary (two-sided) limit mein $x$ kisi bhi side se aa sakta hai. Ek one-sided limit deliberately ek direction block karta hai aur dekhta hai kya hota hai sirf doosri direction use karke. Yahi akela restriction is topic ka poora naya idea hai.

4. Chhote $+$ aur $-$ superscripts — direction of approach

Ab hum jaante hain ki ek one-sided limit kya hai, yahan woh notation hai jo sab ko pehli baar trip karati hai.

Figure 3 ek ruler par dono approaches dikhata hai. Blue dots $a$ ki taraf left se creep kar rahe hain (yeh $a^{-}$ inputs hain, sab $a$ se chhote); orange dots right se creep kar rahe hain ($a^{+}$ inputs, sab $a$ se bade). Dekho arrows red target ki taraf har side se inward point kar rahe hain.

YEH TOPIC ISKO KYU CHAHIYE: yeh do superscripts woh tarika hain jisse hum Section 3 ke one-sided idea ko likhte hain. Yeh "approach" ko do labelled directions mein split karte hain.

5. Word "limit" aur symbol $\lim$

One-sided versions mein sirf direction superscript swap karo:

YEH TOPIC ISKO KYU CHAHIYE: $\lim$ poore chapter ka noun hai. Baaki sab — continuity, derivatives, asymptotes — is word se bana hai. Deeper machinery ke liye Limit of a function — intuitive & ε-δ definition dekho.

6. Symbol $L$ — approached value

• $L$ ::: woh single height value jiske around outputs $f(x)$ pile up karte hain jaise $x\to a$.

YEH TOPIC ISKO KYU CHAHIYE: topic ka punchline yahi hai ki "two-sided limit exist karta hai iff dono sides same $L$ par aim karti hain." "Approached value" ke naam ke bina, tum woh rule state nahi kar sakte.

7. Word "iff" aur symbol $⟺$

Parent ka boxed rule ise use karta hai: ${\lim }_{x\to a}f(x)=L⟺{\lim }_{x\to a^{-}}f(x)={\lim }_{x\to a^{+}}f(x)=L$

Yeh kehta hai: "two-sided limit $L$ ke equal hai" bilkul wahi statement hai jaise "dono one-sided limits exist karte hain aur dono $L$ ke equal hain." Sirf ek direction nahi — ek two-way street.

YEH TOPIC ISKO KYU CHAHIYE: yeh two-sided limit ke poore test ko ek honest logical equivalence mein package karta hai. Yeh topic ka headline theorem hai.

8. Absolute value $|x|$ aur $\infty$ — examples mein do guests

Parent ke worked examples quietly do aur symbols use karte hain. Inhe earn karte hain.

Figure 4 $|x|$ idea ko kaam mein laata hai: yeh sign function $f(x)=|x|/x$ graph karta hai. Dekho blue level zero ke left ki taraf har $x$ ke liye $-1$ par baithe hai, aur orange level right ki taraf $+1$ par. $x=0$ par do hollow dots dikhate hain ki function wahan actually land nahi karta — aur kyunki dono sides alag-alag heights par hain, yeh hamari "left ≠ right" ki sabse clean picture hai.

YEH TOPIC INKO KYU CHAHIYE: $|x|$ ek left≠right mismatch ka sabse clean example build karta hai, aur $\infty$ naam karta hai kya hota hai ek pole par. Dono parent ke examples mein aate hain, isliye dono zero-to-hero clear hone chahiye.

9. Piecewise definition — bada curly brace $\{$

YEH TOPIC ISKO KYU CHAHIYE: dono sides ko disagree karte dekhne ki sabse clean jagah woh hai jahan rule khud switch karta hai. Piecewise notation woh tarika hai jisse hum aisi functions likhte hain.

Prerequisite map

Yahan sab nine foundations topic mein kaise feed karte hain. Diagram top-down padho: top ke paas boxes raw ingredients hain (number line, height machine), middle boxes unhe combine karte hain (approach, direction superscripts, word limit), aur bottom boxes payoff hain (one-sided limits, two-sided rule, worked examples). Har arrow ka matlab "arrow ke peeche wala box arrow ke aage wale box se pehle chahiye."

Equipment checklist

Khud ko test karo — right side cover karo aur reveal karne se pehle answer do.

Connections

• Limit of a function — intuitive & ε-δ definition — $\lim$ ki parent machinery.
• Continuity at a point — ${\lim }_{x\to a}f=f(a)$ chahiye, in symbols se bana.
• Jump, removable & infinite discontinuities — mismatched sides kya cause karti hain.
• Vertical asymptotes — jahan $\infty$ aata hai.
• Greatest integer / floor function — ek piecewise staircase example.
• Differentiability — left & right derivatives — same left/right idea slopes par apply.