Foundations — Limits and continuity in 2D — path-dependence issue

4.4.2 · D1 · Maths › Multivariable Calculus › Limits and continuity in 2D — path-dependence issue

Yeh page ek toolbox hai. Hum har drawer kholenge, har tool ko plain words mein name karenge, uski picture draw karenge, aur batayenge ki parent topic the path-dependence note ko yeh kyun chahiye. Upar se neeche padho — har item sirf usse upar wale items se banta hai.

1. Plane mein point: $(x,y)$

Picture: burnt-orange dot $(x,y)$ par baitha hai. Horizontal axis pe $x$ units chalo, phir $y$ units upar. Ek pair = ek location.

Topic ko yeh kyun chahiye: pura subject yeh poochhhta hai ki ek machine ka output kya hota hai jab input point $(x,y)$ ek special target point $(a,b)$ ki taraf slide karta hai. Points ko name karne ka tarika na ho toh "sliding toward" ki baat hi nahi ho sakti.

2. Target point $(a,b)$ aur arrow $\to$

Yahan $(a,b)$ ek fixed jagah hai jiske baare mein hum sochh rahe hain (often origin $(0,0)$), jabki $(x,y)$ wandering point hai jo uski taraf close hota ja raha hai.

3. Do inputs wali function: $f(x,y)$

Picture: $f$ ko ek landscape ki height socho flat sheet ke upar. Har ground position $(x,y)$ ki terrain height $f(x,y)$ uske upar float karti hai.

Topic ko yeh kyun chahiye: "limit" ka sawaal yeh hai ki "jab main ground par target ki taraf chalta hoon, kya terrain height ek number par settle karti hai?" Do-input function na ho toh sawaal hi nahi.

4. 2D mein Distance: $\sqrt{(x-a{)}^2+(y-b{)}^2}$

Yeh parent note ka sabse important symbol hai, isliye hum ise dheere se build karte hain.

Formula kaise banta hai — right triangle. $(a,b)$ se $(x,y)$ tak do steps mein jao: pehle horizontally $(x-a)$ ka gap, phir vertically $(y-b)$ ka gap. Yeh do moves ek right triangle ki do legs hain; points ke beech ka straight arrow iska hypotenuse hai.

Pythagoras kehta hai (leg)² + (leg)² = (hypotenuse)². Toh

${\text{distance}}^2=(x-a{)}^2+(y-b{)}^2⟹\text{distance}=\sqrt{(x-a{)}^2+(y-b{)}^2}.$

• Hum har gap ko square karte hain taaki negative gap (left ya down jaana) positive ki tarah count ho — length negative nahi ho sakti.
• Hum end mein square root lete hain squaring ko undo karne ke liye aur honest length units mein wapas aane ke liye.

Topic ko yeh kyun chahiye: limit ki definition kehti hai ki input target ke kisi chhote radius ke andar hona chahiye. Yeh formula hi woh "within distance" hai. (Woh radius item 7 mein name karte hain.)

5. Absolute value: $|\cdot |$ aur candidate value $L$

Topic ko yeh kyun chahiye: $|f(x,y)-L|$ measure karta hai ki output $f(x,y)$ candidate answer $L$ se kitna door hai, chahe overshoot kare ya undershoot. Yeh item 4 ki input-side distance ka output-side twin hai.

6. Open disk — "har direction mein close" ki picture

Picture: ek filled circle (minus uski edge aur exact centre). Har ray, har spiral, har zig-zag $(a,b)$ ki taraf ultimately is disk ke andar rehta hai.

7. Greek dials: $\epsilon$ (epsilon) aur $\delta$ (delta)

Topic ko yeh kyun chahiye: paths sirf limit ko disprove kar sakte hain. Yeh prove karne ke liye ki ek limit exist karti hai tumhe poore disk par $\epsilon$–$\delta$ game jeetnaa hoga. Yeh do symbols referee ki rulebook hain.

8. Slope $m$ aur line $y=b+m(x-a)$

$(a,b)$ se hokar jaane wali line slope $m$ ke saath yeh hai: $y=b+m(x-a)$. Alag $m$ target ki taraf ek alag straight road pick karta hai.

Topic ko yeh kyun chahiye: test karne ke liye sabse saste paths straight lines hain. $m$ ko sab values par sweep karna ek saath infinitely many directions test karta hai — aur agar answer phir bhi $m$ par depend karta hai, limit already dead hai (parent note ka Worked Example 1).

9. Polar coordinates: $r$ aur $\theta$

Picture: deep-teal arrow ki length $r$ hai aur woh angle $\theta$ se swing karta hai. Tip ka horizontal axis par shadow $r\cos \theta$ hai (woh $x$ hai); uski height $r\sin \theta$ hai (woh $y$ hai).

10. Squeeze idea aur bound $\le$

Topic ko yeh kyun chahiye: yeh limit exist karne ko conclude karne ka akela honest tarika hai (parent ka Worked Example 3).

11. Degree / homogeneity ("slope kyun survive hua" ka tool)

Prerequisite map

Equipment checklist

Khud ko test karo — parent note padhne se pehle har sawaal ka jawab de sakna chahiye.

Connections

• Multivariable Calculus — parent chapter jo yeh toolbox feed karta hai
• Epsilon-Delta Definition — poora $\epsilon$–$\delta$ challenge game
• Polar Coordinates — origin approach karne ki $r,\theta$ language
• Squeeze Theorem — existence prove karne ka bound-based tool
• Continuity in 1D — 1D story jisko yeh generalize karta hai
• Partial Derivatives — axis directions ke along limits se banta hai
• Differentiability in 2D — in foundations ki continuity support karti hai