Visual walkthrough — Applications — approximation, evaluating limits

4.3.19 · D2 · Maths › Calculus III — Sequences & Series › Applications — approximation, evaluating limits

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Step 1 — Problem: ek aisa fraction jo evaluate hone se inkaar karta hai

KYA HAI. Hum woh value jaanna chahte hain jis par $\frac{\sin x-x}{x^3}$ pahunchta hai jab $x$ zero ki taraf kheenchta hai.

YEH MUSHKIL KYUN HAI. Seedha $x=0$ plug karo: upar hai $\sin 0-0=0$, neeche hai $0^3=0$. Hume milta hai $\frac{0}{0}$ — ek meaningless symbol, koi number nahi. Yeh kuch bhi ho sakta hai. Upar ka zero aur neeche ka zero dono zero ki taraf daud rahe hain, aur jawab poori tarah is baat par depend karta hai ki kaun pehle pahunchta hai.

PICTURE. Dekho kaise upar wali curve $y=\sin x-x$ aur neeche wali curve $y=x^3$ dono origin mein ghus jaati hain. $0$ ke paas yeh ek doosre se chipki rehti hain — yahi poora rahasya ek image mein hai.

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Step 2 — Woh ek idea jo hum lete hain: har function pehle ek polynomial jaisa hota hai

KYA HAI. $x=0$ ke paas, ek smooth function $f$ ko uski Maclaurin series se replace kiya ja sakta hai — ek infinite polynomial jo $0$ par $f$ ki value aur uske saare slopes se banta hai: $f(x)=\underset{\text{height}}{\underbrace{f(0)}}+\underset{\text{slope piece}}{\underbrace{f^{\prime }(0)x}}+\underset{\text{bend piece}}{\underbrace{\frac{f^{\prime \prime }(0)}{2!}{x}^2}}+\underset{\text{next piece}}{\underbrace{\frac{f^{\prime \prime \prime }(0)}{3!}{x}^3}}+\cdots$ Har term ko label kiya gaya hai woh kya karta hai: constant height set karta hai, $x$-term tilt set karta hai, $x^2$-term curvature set karta hai, aur aise aage bhi.

YEH TOOL KYUN, DOOSRA KYUN NAHI. Hum L'Hôpital's Rule use kar sakte the — upar aur neeche baar baar differentiate karo. Lekin woh kyun chupaata hai jawab woh hai. Series har "approach ki speed" ko saaf saaf rakhti hai: $x$ ki sabse choti power bilkul wohi hai jitni tezi se woh function zero se door jaata hai. Yahi woh information hai jo tug-of-war ko chahiye.

PICTURE. Ek ek term aur jodne se polynomial $\sin x$ ko zyada bade stretch par hudh kar leti hai. Dashed accent curve kali true curve ki taraf bahar kheechti jaati hai.

Recall

$\frac{f^{(n)}(0)}{n!}$ kyun? (coefficient, ek line mein) $x^n$ ko exactly $n$ baar differentiate karne par $n!$ bachta hai; baaki har term $x=0$ par zero ho jaata hai. Toh $f^{(n)}(0)$ recover karne ke liye coefficient mein $\frac{1}{n!}$ hona zaroori hai. Poori derivation Taylor & Maclaurin Series mein hai.

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Step 3 — Numerator ki "zero-se-door-jaane ki speed" padhna

KYA HAI. Upar wale, $\sin x-x$, ko standard series $\sin x=x-\frac{x^3}{3!}+\frac{x^5}{5!}-\cdots$ se expand karo: $\sin x-x=(\underset{\text{cancels}}{\underbrace{x}}-\frac{x^3}{6}+\frac{x^5}{120}-\cdots )-\underset{\text{cancels}}{\underbrace{x}}=-\frac{x^3}{6}+\frac{x^5}{120}-\cdots$ Yahan $\frac{x^3}{6}$ yaani $\frac{x^3}{3!}$ hai (kyunki $3!=6$), aur $\frac{x^5}{120}=\frac{x^5}{5!}$ hai.

$x$ CANCEL KYUN HOTA HAI. $\sin x$ aur $x$ ki origin par same height (0) aur same slope (1) hai — woh pehle order par identically zero se door jaate hain. Subtraction us shared part ko wipe kar deta hai, aur pehli jagah expose karta hai jahan woh differ karte hain: $x^3$ term. Toh numerator zero se $x^3$ ki tarah, "cubic speed" par door jaata hai.

PICTURE. Seedhi line $y=x$ aur curve $y=\sin x$ $0$ par tangent hain; unka gap $\sin x-x$ sirf ek halke $x^3$ dip ki tarah khulta hai.

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Step 4 — Denominator ki "zero-se-door-jaane ki speed" padhna

KYA HAI. Neeche wala pehle se ek pure polynomial hai: $x^3$. Iska leading (aur eka) power $x^3$ hai.

YEH AASAAN SIDE KYUN HAI. Koi expansion nahi chahiye — $x^3$ apni series khud hi hai. Iska zero-se-door-jaane ki speed bilkul cubic hai. Gaur karo ki numerator (Step 3) bhi cubic hai. Yahi equality aane wala punchline hai: ek cubic ka doosre cubic se race dena ek finite, non-zero answer deta hai.

PICTURE. Teen curves stack ki gayi hain: $x$ (zero se sabse dheere door jaata hai, linear), $x^2$ (tezi se), $x^3$ (teeno mein se sabse tezi se). Accent $x^3$ woh hai jo hamare numerator ki degree se match karta hai.

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Step 5 — Shared power cancel karo aur $x$ ko zero ki taraf crawl karne do

KYA HAI. Numerator series ko $x^3$ se divide karo: $\frac{\sin x-x}{x^3}=\frac{-\frac{1}{6}{x}^3+\frac{1}{120}{x}^5-\cdots }{x^3}=\underset{\text{survives}}{\underbrace{-\frac{1}{6}}}+\underset{\to 0}{\underbrace{\frac{1}{120}{x}^2-\cdots }}$ Har term mein kam se kam $x^3$ tha, toh $x^3$ se divide karne par ek constant $-\frac{1}{6}$ bachta hai plus aisi terms jo abhi bhi $x$ carry karti hain.

YEH KHATAM KYUN KARTA HAI. Ab koi $\frac{0}{0}$ nahi hai — fraction ek ordinary polynomial in $x$ ban gaya. $x\to 0$ bheejo: constant $-\frac{1}{6}$ rehta hai, $x$ wali har cheez mar jaati hai. Limit seedha padh lo.

PICTURE. Messy quotient function $x\to 0$ par height $-\frac{1}{6}$ par horizontal accent line par flat ho jaata hai.

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Step 6 — Edge case A: bahut KAM terms rakhna (classic trap)

KYA HAI. Maan lo tumne lazily $\sin x\approx x$ (ek term) use kiya. Tab $\sin x-x\approx 0$, aur tum conclude karte ki limit $\frac{0}{x^3}=0$ hai. Galat.

YEH KYUN FAIL HOTA HAI. Tumne wohi term ($-\frac{x^3}{6}$) throw away kar di jo division ke baad bachti hai. Tumhe numerator ko kam se kam denominator ki degree tak — yahan degree $3$ — expand karna hi padega, warna tum jawab delete kar dete ho.

PICTURE. Ek "term budget" bar: ek term kaafi nahi hai, sach pehli baar $x^3$ term par dikhta hai (accent), isse aage ki sab kuch safely drop ki ja sakti hai.

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Step 7 — Edge case B: twist ke saath tie ($x-\tan x$)

KYA HAI. Parent ka sabse tricky example: $\underset{x\to 0}{\lim }\frac{x-\sin x}{x-\tan x}$. Dono expand karo: $x-\sin x=\frac{x^3}{6}-\cdots ,x-\tan x=x-(x+\frac{x^3}{3}+\cdots )=-\frac{x^3}{3}-\cdots$ Yahan $\tan x=x+\frac{x^3}{3}+\cdots$ hai, toh $x$ mein se subtract karne par $-\frac{x^3}{3}$ bachta hai.

DONO CUBIC KYUN HAIN. $\sin x$ $x$ se neeche curve karta hai (neeche dip karta hai $\Rightarrow$ minus flip ke baad positive $\frac{x^3}{6}$), $\tan x$ $x$ se upar curve karta hai (leading $-\frac{x^3}{3}$). Dono cubic speed par zero se door jaate hain lekin opposite signs ke saath — woh sign hi final answer ko negative banata hai. Leading coefficients ka ratio: $\frac{x^3/6}{-x^3/3}=\frac{1/6}{-1/3}=-\frac{1}{2}.$

PICTURE. $0$ ke aas paas: $\tan x$ line $y=x$ ke upar chadha rehta hai, $\sin x$ uske neeche rehta hai. Dono gaps (accent) opposite directions mein point karte hain — woh opposition hi answer mein minus sign hai.

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Step 8 — Edge case C: jab race tie NAHI hoti

KYA HAI. Doosre do outcomes dekhne ke liye denominator badlo:

• $\underset{x\to 0}{\lim }\frac{\sin x-x}{x^2}=\underset{x\to 0}{\lim }\frac{-x^3/6+\cdots }{x^2}=\underset{x\to 0}{\lim }(-\frac{x}{6}+\cdots )=0.$ Numerator (cubic) denominator (quadratic) ko beat karta hai $\Rightarrow$ $0$ par collapse.
• $\underset{x\to 0}{\lim }\frac{\sin x-x}{x^4}=\underset{x\to 0}{\lim }\frac{-x^3/6+\cdots }{x^4}=\underset{x\to 0}{\lim }(-\frac{1}{6x}+\cdots )=\mp \infty .$ Denominator (quartic) jeetta hai $\Rightarrow$ blow up.

YEH KYUN DIKHATE HAIN. Taaki tum kabhi guess na karo. Degree comparison pehle se hi — kisi bhi arithmetic se pehle — decide karta hai ki tumhe $0$, ek number, ya infinity milegi. Sirf degree-match ($m=k$) finite non-zero value deta hai.

PICTURE. Numerator ki cubic ko denominators $x^2,x^3,x^4$ ke against teen panels: accent column mark karta hai kaunsa tie karta hai.

Recall Blow-up ka sign (

$x^4$ denominator) Jab $x\to 0^{+}$, $x^4>0$ hai, toh $-\frac{1}{6x^4}\cdot x^3=-\frac{1}{6x}\to -\infty$. Jab $x\to 0^{-}$, yeh $\to +\infty$. Do-sided limit exist nahi karti — yeh batane laayak hai, kyunki ek akela "$\infty$" chupaata hai ki dono sides disagree kar rahi hain.

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Ek-picture summary

KYA HAI. Sab kuch compress karke: numerator aur denominator ko har ek $x$ ki sabse choti surviving power ke hisaab se rank kiya jaata hai; limit un dono ranks ko compare karke decide hoti hai, aur jab woh match karte hain toh leading coefficients ka ratio hoti hai.

Recall Feynman retelling — poora walkthrough saadhe alfaazon mein

Ek $\frac{0}{0}$ fraction ek race hai: upar aur neeche dono zero ki taraf daude jaate hain, aur hum poochte hain kaun tezi se jaata hai. Speeds measure karne ke liye hum har curve ko uske polynomial twin se swap karte hain — twin ki sabse choti power of $x$ batati hai ki woh zero se kitni tezi se door jaata hai. $\sin x-x$ ke liye linear parts cancel ho jaate hain (sine aur line $x$ pehle order par identically zero se door jaate hain), toh upar wala sirf $x^3$ term par differ karta hai: cubic speed. Neeche wala $x^3$ bhi cubic hai. Same speed = tie, aur tie hamesha ek clean number deta hai — sirf leading coefficients ka ratio, $-\frac{1}{6}÷1=-\frac{1}{6}$. Agar upar wala tezi hota toh $0$ milta; agar neeche wala tezi hota, toh infinity. Haarne ka eka tarika hai bahut kam terms rakhna aur accidentally woh term erase karna jo race decide karta hai — toh hamesha har series ko denominator se ek power aage tak carry karo, phir cancel karo aur $x$ ko zero ki taraf crawl karne do.

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Connections

• Parent topic
• Taylor & Maclaurin Series
• Power Series — Operations (add, multiply, differentiate)
• L'Hôpital's Rule
• Alternating Series Test
• Radius & Interval of Convergence
• Lagrange Remainder & Error Estimation

Concept Map