L'Hôpital's rule — proof using linear approximation, 0 - 0, ∞ - ∞, other indeterminate forms: Visual walkthrough

4.1.26 · D2 · Maths › Calculus I — Limits & Derivatives › L'Hôpital's rule — proof using linear approximation, 0 - 0,

Yeh main note ka companion hai. Yahan tasveerein argument ko aage le jaati hain.

Step 1 — $\frac{0}{0}$ dikhta kaisa hai?

KYA HAI. Hamare paas do functions hain. Upar wale ko $f$ aur neeche wale ko $g$ kaho. Hum chahte hain ki $\frac{f ( x )}{g ( x )}$ ki value kya hogi jab $x$ ek special number $a$ ki taraf khisak raha ho. Symbol $x \to a$ ka matlab sirf itna hai ki " $x$ aur aur $a$ ke paas jaata rehta hai". Mushkil yeh hai: $x = a$ par dono functions zero ke barabar hain.

YEH PROBLEM KYUN HAI. Zero divided by zero koi number nahi hai — yeh ek sawaal hai. Tasveer dekho: do alag-alag curves dono ek hi jagah zero se guzar sakti hain, phir bhi us jagah ke paas unka ratio kuch bhi ho sakta hai. Sirf form dekh ke kuch nahi pata chalta.

TASVEER. Do curves, $f$ (blue) aur $g$ (orange), dono $x = a$ par horizontal axis ko cross karti hain. Red dot shared zero ko mark karta hai.

Figure — L'Hôpital's rule — proof using linear approximation, 0 - 0, ∞ - ∞, other indeterminate forms

Step 2 — Zoom karo: curve hi uski tangent line hai

KYA HAI. Kisi bhi curve ko lo aur apna camera $x = a$ point par hard zoom karo. Jaise-jaise tum zoom karte ho, curve ghumi-phiri dikhna band ho jaati hai aur ek seedhi line jaisi lagti hai — uski tangent line. Yeh Linear Approximation & Tangent Lines ka idea hai.

YEH TOOL KYUN, KUCH AUR KYUN NAHI? Humein ek pechida curve ki jagah koi simple cheez chahiye. Tangent line us point par curve ki sabse behtar straight-line copy hai — yeh height aur slope dono match karti hai. Aur kisi jaane-pehchane point se guzarne wali seedhi line likhna bilkul aasaan hai. Yahi woh tool hai jo "mushkil curve ratio" ko "aasaan line ratio" mein badal deta hai.

TASVEER. Left par blue curve tedhi lagti hai. Right par, ek zoomed box same curve ko uski tangent line (dashed) ke saath almost perfectly overlap karta dikhata hai. Green segment slope dikhata hai.

Same copy $g$ ke liye bhi kaam karti hai: $g (x) \approx g (a) + g^{'} (a) (x - a) .$

Step 3 — Constant terms ko khatam karo

KYA HAI. Apni do tangent-line copies ko ratio mein substitute karo, phir Step 1 ki baat use karo ki $f (a) = 0$ aur $g (a) = 0$ .

KYUN. Constant terms $f (a)$ aur $g (a)$ $a$ par heights hain. Yahan dono zero hain — yahi " $\frac{0}{0}$ " ka poora matlab hai. Toh yeh seedhe gayab ho jaate hain, aur jo bachta hai woh sirf slope times step hai.

TASVEER. Dono tangent lines ab origin-height (axis) se guzarti hain. Unki poori kahaani ab sirf unki slopes hain — maano blue line orange se zyada steep hai.

$\frac{f ( x )}{g ( x )} \approx \frac{f ( a ) = 0 + f ^{'} ( a ) ( x - a )}{= 0 g ( a ) + g ^{'} ( a ) ( x - a )} = \frac{f ^{'} ( a ) ( x - a )}{g ^{'} ( a ) ( x - a )}$

Struck-out $f (a), g (a)$ ::: chale gaye, kyunki dono $a$ par zero hain.
Jo bacha ::: slope times wahi ek chhota step, upar aur neeche.

Step 4 — Shared smallness cancel karo

KYA HAI. Factor $(x - a)$ upar aur neeche dono jagah hai. Yeh dono mein wahi chhoti si number hai, toh cancel ho jaati hai.

YEH PUNCHLINE KYUN HAI. $(x - a)$ hi shared shrinking hai — yeh "dono zero ki taraf race karna" wala hissa hai. Jab tum dekho ki yeh upar aur neeche ek jaisa hai, toh yeh answer decide nahi kar sakta. Ise hatao aur jo akela khada rehta hai woh sirf slopes ka ratio hai — speeds ka ratio.

TASVEER. Upar aur neeche $(x - a)$ ko same color mein highlight kiya gaya, dono par slash — sirf $f^{'} (a) / g^{'} (a)$ bachta hai.

$\frac{f ^{'} ( a ) ( x - a )}{g ^{'} ( a ) ( x - a )} = \frac{f ^{'} ( a )}{g ^{'} ( a )}$

Step 5 — Limit sach mein lo

KYA HAI. Upar sab kuch " $\approx$ " use kiya. Ab $x$ ko actually $a$ mein slide karne do. Jaise-jaise hota hai, tangent-line copies exact ho jaati hain (unka gap $(x - a)$ se tezi se mita), toh approximation equality ban jaati hai.

KYUN. Hum limit chahte the, approximation nahi. Limit woh jagah hai jahan tangent-line copy aur asli curve perfectly agree karti hain.

TASVEER. $x$ ki ek ke baad ek closer values $a$ ki taraf march karti hain; ratio ki value height $f^{'} (a) / g^{'} (a)$ par horizontal dashed line par settle hoti hai.

Step 6 — Honest fix: Cauchy's Mean Value Theorem

KYA HAI. Steps 2–5 ne quietly maan liya tha ki slopes $f^{'}, g^{'}$ khud smooth (continuous) hain. Poori tarah airtight hone ke liye hum "slope at $a$ " ko " $a$ aur $x$ ke beech mein kahin slope" se replace karte hain, Cauchy Mean Value Theorem use karke.

KYUN. Cauchy MVT humein ek exact equation deta hai — koi " $\approx$ " nahi kahin bhi — jo kehta hai: $a$ aur $x$ ke beech kahin ek point $c$ hai jahan total changes ka ratio instantaneous slopes ke ratio ke barabar hai.

TASVEER. Axis par $a$ se $x$ tak ka interval, ek point $c$ (green) strictly andar trapped. Ek arrow dikhata hai $c$ $a$ ki taraf squeeze hota jab $x \to a$ .

$\frac{f ( x ) - f ( a ) 0}{g ( x ) - 0 g ( a )} = \frac{f ^{'} ( c )}{g ^{'} ( c )}, c between a and x$

Kyunki $f (a) = g (a) = 0$ hai, left side exactly $\frac{f ( x )}{g ( x )}$ hai. Jab $x \to a$ , trapped point $c$ ke paas jaane ki koi jagah nahi siwaaye $a$ ke, toh $\frac{f ^{'} ( c )}{g ^{'} ( c )} \to lim \frac{f ^{'}}{g ^{'}}$ . Wahi conclusion — koi assumption nahi ki derivatives continuous hain.

Step 7 — Edge case: agar rule hamesha loop kare toh?

KYA HAI. Rule apna answer tabhi deta hai jab right-hand limit $lim f^{'} / g^{'}$ actually exist kare (finite ya $\pm \infty$ ). Kabhi-kabhi differentiate karne se ek nayi fraction milti hai jo purani jaisi buri hoti hai, hamesha ke liye.

YEH KYUN DIKHAYEIN. Jo reader rule par blindly trust karta hai woh chakkar mein phase jaayega. Socho $lim_{x \to \infty} \frac{e ^{x}}{e ^{2 x} + 1} .$ Har application fraction ko ulta kar deta hai aur kabhi simplify nahi hota. Bachne ka raasta algebra hai: upar aur neeche $e^{x}$ se divide karo.

TASVEER. Ek loop arrow (rule apne aap mein cycle karta) ke saath ek seedha arrow (algebra) jo answer $1$ par land karta hai.

$\frac{e ^{x}}{e ^{2 x} + 1} = \frac{1}{1 + e ^{- 2 x}} x \to \infty \frac{1}{1 + 0} = 1$

$e^{- 2 x}$ ::: $x \to \infty$ hone par $0$ mein simaṭ jaata hai (dekho Exponential & Logarithm Growth Rates).
Poori cheez saaf taur par $1$ par settle hoti hai — koi looping nahi.

Step 8 — Edge case: forms jo pehle convert karni hoti hain

KYA HAI. Sirf $\frac{0}{0}$ aur $\frac{\infty}{\infty}$ directly rule ko feed karte hain. Baaki indeterminate forms disguise mein hain — unhe pehle fraction mein reshape karo.

KYUN. L'Hôpital ek ratio ko differentiate karta hai. Agar tumhara problem product $0 \cdot \infty$ ya power $1^{\infty}$ hai, toh abhi koi ratio nahi hai — tumhe ek banana hoga.

TASVEER. Ek chhota flow map: har stray form $\frac{0}{0}$ ya $\frac{\infty}{\infty}$ mein arrow kar raha hai.

Ek-tasveer summary

Yeh single figure poora safar compress karta hai: ek shared zero se do curves (Step 1) → har ek apni tangent line se replace (Step 2) → constants mar jaate hain (Step 3) → shared $(x - a)$ cancel hota hai (Step 4) → limit slope ratio ban jaati hai (Step 5).

Recall Feynman retelling — seedha bolo

Do curves dono ek hi jagah zero mein gir jaati hain, toh unka ratio uss pareshan karne wale $\frac{0}{0}$ mein ban jaata hai. Do gayab hoti heights ko ghoorte rehne ki bajaye, main zoom in karta hoon jab tak har curve apni seedhi tangent line nahi ban jaati — ek aisi line jo main sirf apni slope aur ek step $(x - a)$ se likh sakta hoon. Kyunki dono height zero se shuru hote hain, constant terms gayab ho jaate hain; dono lines sirf slope times same chhota step hain. Woh chhota step $(x - a)$ upar aur neeche ek jaisa hai, toh cancel ho jaata hai — woh "dono zero ki taraf race" wala hissa tha aur winner decide nahi kar sakta tha. Jo bachta hai woh sirf slopes ka ratio hai, $f^{'} (a) / g^{'} (a)$ : speeds ka ratio. Cauchy ka theorem ise slopes ko ek point $c$ par pin karke exact bana deta hai, jo $a$ ki taraf squeeze hota hai. Aur main imandaar rehta hoon: trick tabhi payoff deti hai jab woh speed-ratio actually settle ho jaaye; agar loop kare, main algebra ki taraf jaata hoon.

Connections

Linear Approximation & Tangent Lines — Steps 2–4 literally yahi idea action mein hain.
Mean Value Theorem — Step 6 ka Cauchy MVT tasveer ko rigorous banata hai.
Taylor Series — agla level: sirf pehli slopes nahi, leading nonzero terms compare karo.
Indeterminate Forms — Step 8 ka disguises ka chidiyaghar.
Exponential & Logarithm Growth Rates — Step 7 ke edge case ko power karta hai.
Limits — Definition & Laws — " $lim$ " aur " $x \to a$ " ka actually matlab kya hai.