Limits and continuity in 2D — path-dependence issue: Visual walkthrough

4.4.2 · D2 · Maths › Multivariable Calculus › Limits and continuity in 2D — path-dependence issue

Hum assume karte hain ki tum sirf itna jaante ho: ek point $(x, y)$ ek flat map par ek jagah hai, aur ek function $f (x, y)$ us jagah ko ek height deta hai. Baaki sab — disk, ray, slope, polar trick — hum raaste mein seekhenge.

Step 1 — 2D limit asal mein kya pooch raha hai

KYA. Ek flat map ( $x y$ -plane) imagine karo aur har jagah ke upar ek height $f (x, y)$ . Hum origin ke upar hover karte hain aur poochte hain: kya us ke paas ki height ek fixed value par aati hai?

"Disk" word kyun. 1D mein tum ek point ke paas left road ya right road se aate ho — sirf do tarike. Ek map par tum kisi bhi direction se aur kisi bhi curve ke saath aa sakte ho. "Har taraf se origin ke kareeb" ko honestly capture karne ka tarika ek chhota bhra hua circle hai — ek disk — radius $δ$ ka (ek tiny distance jise hum shrink kar sakte hain). Figure dekho: shaded blue disk origin se $δ$ distance ke andar ke sab points hain. Limit rule demand karta hai ki height poore disk par $L$ ke paas ho, na ki kuch roads par.

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

Yahan $δ$ (Greek letter "delta") sirf ek naam hai "kitna close kaafi close hai" ke liye, aur $x^{2} + y^{2}$ $(x, y)$ se origin tak ki straight-line distance hai — figure mein khicha hua arrow ka length. Full precision version ke liye Epsilon-Delta Definition dekho.

Step 2 — Hamara test function, aur use height ke roop mein padhna

Hum parent ka canonical villain padhte hain: $f (x, y) = spot (x, y) par height \frac{x y}{x ^{2} + y ^{2}} .$

$x y$ (upar wala) positive hota hai jab $x, y$ ka sign same ho, negative jab different ho, zero kisi bhi axis par.
$x^{2} + y^{2}$ (neeche wala) origin tak ki squared distance hai — hamesha positive, siwa origin ke, jahan woh $0$ hai (toh $f$ exactly us point par undefined hai jis ki taraf hum limit le rahe hain — isliye yeh interesting hai).

KYA. Figure map ko height se colour karta hai: warm jahan $f > 0$ , cool jahan $f < 0$ .

KYUN. Kisi bhi algebra se pehle, colouring ko dekhna hi answer whisper kar deta hai. Notice karo ki colour kabhi ek shade mein fade nahi hoti jab tum centre ki taraf spiral karte ho — origin ke paas warm aur cool packed saath milte hain. Ek well-behaved limit ek uniform colour dikhata jो centre ko swallow kar le. Yeh nahi dikha raha.

Step 3 — Sabse sasta paths: do axes

KYA. $x$ -axis ke saath andar chalo. Us road par $y = 0$ , toh $f (x, 0) = \frac{x \cdot 0 y}{x ^{2} + y ^{2} 0} = \frac{0}{x ^{2}} = 0.$ Walk ka har step height $0$ read karta hai; is road ke saath limit $0$ hai.

Ab $y$ -axis, jahan $x = 0$ : $f (0, y) = \frac{0 x \cdot y}{x ^{2} 0 + y ^{2}} = \frac{0}{y ^{2}} = 0.$ Yeh bhi $0$ .

Pehle yeh kyun. Yeh sabse sasti substitutions hain — ek variable zero ho jaata hai aur poori cheez ek 1D calculation mein collapse ho jaati hai. Picture dono axis-roads ko height $0$ par glowing dikhati hai centre tak.

Step 4 — Ek saath saari slanted lines

KYA. Origin ke through slope $m$ wali ek straight road $y = m x$ hai ( $m$ batata hai kitni steep: $m = 0$ flat hai, $m = 1$ $4 5^{\circ}$ diagonal hai). Substitute karo:

=\frac{\overbrace{m\,x^2}^{\text{top}}}{\underbrace{x^2(1+m^2)}_{\text{bottom}}} =\frac{m}{1+m^2}.$$ Cancellation term-by-term padho: top hai $x\cdot mx = m x^2$; bottom factor hota hai $x^2(1+m^2)$ ke roop mein; shared $x^2$ **cancel** ho jaata hai (valid kyunki road par, origin se door, $x\ne 0$). Har $x$ gayab ho jaata hai — us road par height **constant** $\dfrac{m}{1+m^2}$ hai, har distance par same. **KYUN yeh death blow hai.** Result mein **koi $x$ nahi bacha** — sirf $m$. Toh limiting value *poori tarah se direction* $m$ se decide hoti hai. Alag roads, alag constants: | slope $m$ | value $\dfrac{m}{1+m^2}$ | |---|---| | $0$ (axis) | $0$ | | $1$ (diagonal) | $\tfrac12$ | | $-1$ | $-\tfrac12$ | Picture kai roads ko stack karti hai unki constant height ke colour ke saath — alag shades ka ek fan jo origin par milta hai. ![[deepdives/dd-maths-4.4.02-d2-s04.png]] **Verdict.** Flat road $0$ kehta hai, diagonal $\tfrac12$ kehta hai. Do paths, do answers ⟹ **limit exist nahi karta.** Koi ek $L$ ek tiny disk ke andar $0$ aur $\tfrac12$ dono ke paas nahi ho sakta. --- ## Step 5 — Polar coordinates ke saath ek saath har case Line test ne ek baar mein ek direction check kiya. **Polar coordinates** ek hi formula mein *sab* directions check karta hai — yeh woh tool hai jo [[Polar Coordinates]] deta hai, aur yeh sahi tool hai kyunki hamara function sirf direction ki parwah karta hai. **KYA.** Point ko uski origin se $r$ distance aur positive $x$-axis se measured angle $\theta$ (theta) se likho: $$x=r\cos\theta,\qquad y=r\sin\theta,\qquad x^2+y^2=r^2.$$ Origin ke paas aana ab simply $r\to 0$ hai, kisi bhi fixed heading $\theta$ par. Substitute karo: $$f=\frac{(r\cos\theta)(r\sin\theta)}{r^2} =\frac{r^2\cos\theta\sin\theta}{r^2} =\cos\theta\sin\theta.$$ Term-by-term: top hai $r^2\cos\theta\sin\theta$, bottom hai $r^2$, $r^2$ cancel ho jaata hai, aur **distance $r$ poori tarah gone hai.** Jo bacha, $\cos\theta\sin\theta$, woh *sirf heading $\theta$ par* depend karta hai. **KYUN yeh hamesha ke liye settle kar deta hai.** Agar $r\to 0$ ke baad bhi answer mein $\theta$ hai, toh woh bachta hua $\theta$ **hi** path-dependence hai, algebra mein. Yahan $\cos\theta\sin\theta$ $-\tfrac12$ (at $\theta=135^\circ$) se $+\tfrac12$ (at $\theta=45^\circ$) tak range karta hai aur axes par $0$ hai — exactly wahi jo humne har line mein try kiya, ek shot mein saare quadrants cover. Figure $\cos\theta\sin\theta$ ko poore circle ke around plot karta hai: ek curve jo ek flat number hone se mana kar deta hai. ![[deepdives/dd-maths-4.4.02-d2-s05.png]] > [!formula] Polar verdict rule > $x=r\cos\theta,\,y=r\sin\theta$ substitute karne ke baad aur $r\to0$ let karne ke baad: > - **Koi $\theta$ survive nahi karta** (ya $|f-L|\le g(r)$ jahan $g(r)\to0$ *sab* $\theta$ ke liye) ⟹ limit exists. > - **$\theta$ survive karta hai** ⟹ direction matter karta hai ⟹ limit **exist nahi karta**. --- ## Step 6 — Degenerate cases jo tumhe skip nahi karni chahiye > [!intuition] Har corner cover karo > Ek walkthrough tabhi honest hai jab woh weird inputs ka naam le. Unme se teen: **(a) Origin khud.** $(0,0)$ par bottom $x^2+y^2=0$, toh $f$ wahan *undefined* hai — division by zero. Theek hai: limit un points ke baare mein poochta hai jo origin ke *paas* hain, kabhi origin nahi ($0<\sqrt{x^2+y^2}$ definition mein, strict $0<$). Hum $(0,0)$ kabhi plug nahi karte. **(b) Axes as edge directions.** Axes par $xy=0$ toh $f=0$, aur polar mein $\theta\in\{0,90^\circ,180^\circ,270^\circ\}$ deta hai $\cos\theta\sin\theta=0$. Yeh *sirf* woh directions hain jo $0$ read karte hain — yeh special seam hai, rule nahi. **(c) Saare char quadrants.** $\cos\theta\sin\theta$ Quadrants I aur III mein $+$ hai (jahan $x,y$ ka sign same ho) aur Quadrants II aur IV mein $-$. Neeche wali picture plane ko sign se tile karti hai taaki tum centre par milte highs aur lows ka checkerboard dekh sako — ek doomed limit ka visual signature. ![[deepdives/dd-maths-4.4.02-d2-s06.png]] --- ## Step 7 — Contrast: ek **surviving** limit kaisa dikhta hai Failure clearly dekhne ke liye, ek nearby function ko *succeed* hote dekho: $\dfrac{x^2y}{x^2+y^2}$ (parent ka Example 3). Polar mein, $$\frac{(r\cos\theta)^2(r\sin\theta)}{r^2} =\frac{r^3\cos^2\theta\sin\theta}{r^2} =r\cdot\cos^2\theta\sin\theta.$$ Key difference: ek $r$ **aage survive karta hai.** Kyunki $|\cos^2\theta\sin\theta|\le 1$, $$|f|\le r\xrightarrow[r\to0]{}0\quad\text{har }\theta\text{ ke liye}.$$ $\theta$-part ek aisi factor ke andar trap hai jo $r$ ke saath shrink hoti hai, direction chahe koi bhi ho — yeh hai [[Squeeze Theorem]] *poori disk* par kaam karta hua. Figure doomed function (koi $r$ nahi, flat coloured wedges) ko healthy wale ke saath overlay karta hai (aage ek $r$ hai, jiska poora surface $0$ par funnel karta hai). ![[deepdives/dd-maths-4.4.02-d2-s07.png]] > [!recall]- Khud se sawaal karo > Kyun ek $r$ aage limit bachata hai lekin akela $\theta$ use doom karta hai? ::: $r$ kisi bhi direction se $0$ par shrink karta hai, isliye ek shrinking wall (Squeeze) har path ko $0$ par trap karta hai. Akele $\theta$ mein koi shrinking wall nahi hoti — alag headings alag constants par lock ho jaati hain. > $\frac{xy}{x^2+y^2}$ kahan undefined hai, aur kya woh limit ko tod deta hai? ::: Sirf $(0,0)$ par; yeh limit definition ko nahi todta, jo centre point khud ko exclude karta hai. --- ## Ek-picture summary ![[deepdives/dd-maths-4.4.02-d2-s08.png]] Ek image, poora argument: axis roads $0$ read kar rahe hain, diagonal $\tfrac12$ read kar raha hai, values $\cos\theta\sin\theta$ ka polar wheel $-\tfrac12$ aur $+\tfrac12$ ke beech swing kar raha hai, aur verdict banner — **paths disagree ⟹ limit DNE.** > [!recall]- Feynman retelling — poori walk simple shabdon mein > Tum ek foggy field mein ek lamppost ki taraf chal rahe ho, aur "ground ki height" hamara function hai. Pehle tum check karte ho: kya ground *post par* define bhi hai? Nahi — bilkul centre par ek bottomless hole hai (divide by zero ho jaata), lekin theek hai, hum sirf us ke *aaas-paas* ki ground ki parwah karte hain. Tum flat east road se andar chalo: ground poore raaste $0$ rehti hai. North road se chalo: bhi $0$. Tempting hai "ground smooth hai, height $0$!" declare karna — lekin tumne sirf do roads try kiye. Toh tum $45^\circ$ diagonal try karte ho, aur suddenly ground us poori road par height $\tfrac12$ par baithi hai. Do dost do roads se aake do alag heights dekh rahe hain post ke bilkul paas — ground mein wahan ek *crack* hai. Har direction ek saath check karne ke liye tum "kitna door, kaunsi taraf" ($r$ aur $\theta$) par switch karte ho: algebra $\cos\theta\sin\theta$ spurt karta hai — "kitna door" wala $r$ clean cancel ho gaya, sirf "kaunsi taraf" wala $\theta$ bacha. Woh surviving direction-word *hi* crack hai. Ek friendlier post se compare karo jahan algebra aage ek $r$ chhodta hai: jaise tum close aate ho, $r$ *har* direction se zero par shrink karta hai, isliye sab height $0$ par agree karte hain — woh limit jeeta hai. **Ek lesson: agar distance cancel ho jaaye aur direction survive kare, toh limit dead hai.** > [!mnemonic] Distance cancels, Direction survives → DNE. Aur: *Lines Lie, the Wheel (polar) Tells All.* --- ## Connections - [[Multivariable Calculus|Limits and continuity in 2D — path-dependence issue]] — parent topic - [[Polar Coordinates]] — $r,\theta$ substitution jo ek saath sab directions check karta hai - [[Squeeze Theorem]] — shrinking-$r$ wall jo *prove* karta hai ki limit exist karta hai - [[Epsilon-Delta Definition]] — "close" ka disk-based precise matlab - [[Continuity in 1D]] — two-road world jis se hum generalise kar rahe hain - [[Partial Derivatives]] — axis-directions ke saath liye gaye limits - [[Differentiability in 2D]] — continuity chahiye, jo path-independence maangta hai