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

4.1.26 · Maths › Calculus I — Limits & Derivatives

Problem KYA hai?

Jab aap $x \to a$ ko $\frac{f ( x )}{g ( x )}$ mein plug karte hain aur $\frac{0}{0}$ ya $\frac{\infty}{\infty}$ milta hai, toh expression indeterminate hai — value sirf form se determine nahi hoti. Compare karo:

$lim_{x \to 0} \frac{s i n x}{x} = 1, lim_{x \to 0} \frac{s i n x}{x ^{2}} = \infty, lim_{x \to 0} \frac{x ^{2}}{s i n x} = 0.$

Teeno $\frac{0}{0}$ jaisi lagti hain phir bhi bilkul alag answers deti hain. Toh $\frac{0}{0}$ ek sawaal hai, jawaab nahi.

YEH SACH KYO hai — linear approximation se derivation

Yeh "Feynman / derivation-from-scratch" wala view hai. $\frac{0}{0}$ case lo jahan $f (a) = g (a) = 0$ ho (continuity se define karo unhe).

Step 1 — Near $a$ linearise karo. Kisi point ke paas, ek differentiable function apni tangent line jaisi dikhti hai: $f (x) \approx f (a) + f^{'} (a) (x - a), g (x) \approx g (a) + g^{'} (a) (x - a) .$ Yeh step kyun? Derivatives ki poori power yahi hai ki curve apni tangent line hai first order par — error $(x - a)$ se bhi tezi se vanish ho jaati hai.

Step 2 — $f (a) = g (a) = 0$ use karo. Constant terms mar jaate hain: $\frac{f ( x )}{g ( x )} \approx \frac{f ^{'} ( a ) ( x - a )}{g ^{'} ( a ) ( x - a )} = \frac{f ^{'} ( a )}{g ^{'} ( a )} .$ Yeh step kyun? $(x - a)$ factors shared "smallness" hain. Yeh cancel ho jaate hain, ratio of speeds bach jaata hai.

Step 3 — Limit lo. Jaise $x \to a$ hota hai, approximation exact ho jaata hai: $lim_{x \to a} \frac{f ( x )}{g ( x )} = \frac{f ^{'} ( a )}{g ^{'} ( a )} = lim_{x \to a} \frac{f ^{'} ( x )}{g ^{'} ( x )} .$

Rigorous version (Cauchy Mean Value Theorem)

Linear-approximation argument secretly assume karta hai ki $f^{'}, g^{'}$ continuous hain. Airtight proof Cauchy MVT use karta hai: $f, g$ ke liye $[a, x]$ par ek $c \in (a, x)$ hota hai jahan $\frac{f ( x ) - f ( a )}{g ( x ) - g ( a )} = \frac{f ^{'} ( c )}{g ^{'} ( c )} .$ $f (a) = g (a) = 0$ ke saath yeh $\frac{f ( x )}{g ( x )} = \frac{f ^{'} ( c )}{g ^{'} ( c )}$ ban jaata hai. Jaise $x \to a$ hota hai, $c$ squeeze hokar $a$ par aa jaata hai, toh limits match karti hain. Yeh kyun stronger hai: derivatives ki continuity ki zaroorat nahi.

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

KAISE use karein — aur indeterminate-form zoo

7 indeterminate forms hoti hain. Sirf $\frac{0}{0}$ aur $\frac{\infty}{\infty}$ L'Hôpital mein directly jaati hain; baaki ko pehle convert karna padta hai.

Form	Convert karne ki trick	Result type
$0/0$	directly use karo	—
$\infty/\infty$	directly use karo	—
$0 \cdot \infty$	likho $f \cdot g = \frac{f}{1/ g}$	$0/0$ ya $\infty/\infty$
$\infty - \infty$	common denominator / factor	$0/0$
$0^{0}, \infty^{0}, 1^{\infty}$	$ln$ lo: $y = f^{g} \Rightarrow ln y = g ln f$	$0 \cdot \infty$

Worked examples

Common mistakes (Steel-man karo inhe)

Active recall

Recall Quick self-test (answers cover karo)

Kon si do forms par directly apply kar sakte ho? ::: $\frac{0}{0}$ aur $\frac{\infty}{\infty}$ .
Kya tum quotient rule use karte ho? ::: Nahi — numerator aur denominator ko alag-alag differentiate karo.
$0^{0}$ ko kaise handle karte ho? ::: $ln$ lo, $0 \cdot \infty$ mein convert karo, phir $0/0$ mein.
Conclusion hold karne ke liye kya true hona chahiye? ::: $f^{'} / g^{'}$ ki limit exist karni chahiye (ya $\pm \infty$ honi chahiye).

Recall Feynman: 12-saal ke bacche ko explain karo

Do ghonge ek hi line se start karte hain aur hum poochhte hain "thodi der baad kaun aage hoga?" Kyunki dono ne zero se start kiya, sirf ek cheez decide karti hai — kaun kitna tezi se chhalta hai — wahi slope hai, derivative hai. Toh unke tiny distances compare karne ki jagah (jo dono practically zero hain — confusing!), hum sirf unki speeds compare karte hain. Yahi swap, distances → speeds, L'Hôpital's rule hai.

Flashcards

State L'Hôpital's rule for

0/0

Agar

f (a) = g (a) = 0

g^{'} \neq = 0

near

a

, aur

lim f^{'} / g^{'}

exist karta hai, tab

lim_{x \to a} f / g = lim_{x \to a} f^{'} / g^{'}

Why does linear approximation explain the rule?

Near

a

f \approx f^{'} (a) (x - a)

aur

g \approx g^{'} (a) (x - a)

;

(x - a)

cancel ho jaate hain aur

f^{'} (a) / g^{'} (a)

bach jaata hai.

Which theorem makes the proof rigorous?

The Cauchy Mean Value Theorem.

List the 7 indeterminate forms.

0/0, \infty/\infty, 0 \cdot \infty, \infty - \infty, 0^{0}, \infty^{0}, 1^{\infty}

How do you convert

0 \cdot \infty

Likho

f \cdot g = f / (1/ g)

taaki

0/0

\infty/\infty

ban jaaye.

How do you handle

1^{\infty}, 0^{0}, \infty^{0}

Natural log lo taaki

0 \cdot \infty

mile, fraction mein convert karo, rule apply karo, phir exponentiate karo.

Common error: applying which rule by mistake?

Quotient rule — uski jagah top aur bottom ko independently differentiate karo.

lim_{x \to 0} (e^{x} - 1) / x = ?

1

lim_{x \to \infty} (ln x) / x = ?

0

lim_{x \to \infty} (1 + 1/ x)^{x} = ?

e

lim f^{'} / g^{'}

does not exist, what can you conclude about

lim f / g

Kuch nahi — rule one-directional hai;

lim f / g

phir bhi exist kar sakta hai.

Connections

Mean Value Theorem — Cauchy MVT proof ka engine hai.
Linear Approximation & Tangent Lines — intuitive derivation.
Taylor Series — generalise karta hai: leading nonzero Taylor terms ka ratio limit deta hai.
Indeterminate Forms — broader classification.
Limits — Definition & Laws — limit ko kya satisfy karna chahiye.
Exponential & Logarithm Growth Rates — example (3) growth speeds compare karta hai.