Important limits — lim(sin x - x) = 1, lim((1+1 - n)ⁿ) = e: Worked examples

4.1.6 · D3 · Maths › Calculus I — Limits & Derivatives › Important limits — lim(sin x - x) = 1, lim((1+1 - n)ⁿ) = e

Scenario matrix

In do limits par bane har problem neeche diye gaye cells mein se kisi ek mein aata hai. Right column us example ka naam batata hai jo us cell mein aata hai.

#	Cell (kya cheez ise different banati hai)	Kaunsa limit	Chhupi hui trap	Example
1	$\frac{s i n ( scaled )}{scaled}$ , plain rescale	sin	arguments match karna bhool jaana	A
2	Higher power $x^{3}$ , needs the $\frac{1 - c o s x}{x ^{2}}$ helper	sin	power ko galat split karna	B
3	Two-sided check: does the limit exist as $x \to 0^{-}$ too? ( $x$ ka sign)	sin	even/odd blindly assume karna	C
4	Geometric / word problem (arc vs chord shrinking)	sin	picture galat padhna	D
5	Degrees, radians nahi (the " $= 1$ is a lie" case)	sin	degrees mein radian value use karna	E
6	$(1 + \frac{a}{x})^{x}$ negative $a$ ke saath	$e$	exponent ka sign	F
7	Base $\frac{1}{x}$ ki tarah nahi shrink karta (degenerate $e$ )	$e$	pattern bahut jaldi match karna	G
8	Real-world compound growth (continuous interest)	$e$	galat $a$ choose karna	H
9	Exam twist: ek expression mein dono limits mix hain	both	pieces ko alag evaluate karna jab nahi kar sakte	I

Rows 1–5 sin limit ko sign, power, dimension, aur units ke across stress karte hain. Rows 6–8 $e$ limit ko sign aur degeneracy ke across stress karte hain. Row 9 boss fight hai.

Shuru karne se pehle, do engines jo hum baar baar reuse karte hain. Neeche sab kuch inhi mein se ek hai, disguise mein:

Example A — plain rescale (cell 1)

Worked example Compute karo

$x \to 0 lim \frac{sin 7 x}{sin 2 x}$ .

Forecast: padhne se pehle answer guess karo. (Hint: dono apne arguments ki tarah shrink karte hain — to yeh arguments ka ratio hai.)

Step 1. Multiply aur divide karo taaki har sin ko uske apne argument ke saath force kar sako: $\frac{s i n 7 x}{s i n 2 x} = \frac{s i n 7 x}{7 x} \cdot \frac{2 x}{s i n 2 x} \cdot \frac{7 x}{2 x} .$ Yeh step kyun? Master $\frac{s i n u}{u}$ ko upar aur neeche argument match karna chahiye. Yahan do alag sines hain, to hum har ek ko uska matching denominator dete hain, phir bache hue $\frac{7 x}{2 x}$ se correct karte hain.

Step 2. Jab $x \to 0$ : $7 x \to 0$ to $\frac{s i n 7 x}{7 x} \to 1$ ; similarly $\frac{2 x}{s i n 2 x} \to 1$ ( $\to 1$ quantity ka reciprocal bhi $1$ hota hai). Last factor $\frac{7 x}{2 x} = \frac{7}{2}$ already constant hai. Yeh step kyun? Hum master limit ek baar har sine ke liye use karte hain, phir bachne wale numbers multiply karte hain.

Step 3. $1 \cdot 1 \cdot \frac{7}{2} = \frac{7}{2} .$

Verify: chhota $x = 0.001$ plug karo: $sin (0.007) / sin (0.002) \approx 0.006999.../0.001999... \approx 3.4999 \approx \frac{7}{2}$ . ✓

Example B — higher power, cosine helper chahiye (cell 2)

Example C — $x$ ka sign: kya limit dono sides se exist karti hai? (cell 3)

Worked example Dikhao ki

$x \to 0 lim \frac{sin x}{x} = 1$ left se bhi, yaani $x \to 0^{-}$ par.

Forecast: humne ise $x > 0$ ke liye areas use karke prove kiya. Kya negative jaana answer badalta hai ya tod deta hai?

Step 1. Graph dekho. $x = - t$ set karo jahan $t > 0$ ho. $\frac{s i n ( - t )}{- t} = \frac{- s i n t}{- t} = \frac{s i n t}{t} .$ Yeh step kyun? $sin$ ek odd function hai ( $sin (- t) = - sin t$ ), aur denominator ka sign bhi flip hota hai. Dono minus signs cancel ho jaate hain — ratio even hai.

Step 2. Jab $x \to 0^{-}$ , $t \to 0^{+}$ , aur $\frac{s i n t}{t} \to 1$ already proven right-hand result se. Yeh step kyun? Humne untested left side ko tested right side par reduce kar diya.

Step 3. Left limit $=$ right limit $= 1$ , to two-sided limit exist karti hai aur $1$ ke barabar hai.

Verify: $x = - 0.001$ : $sin (- 0.001) / (- 0.001) = sin (0.001) /0.001 \approx 0.99999983$ . ✓ Positive side jaisa hi.

Example D — geometric word problem: arc vs chord (cell 4)

Worked example Ek point unit circle par ride kar raha hai. Jab central angle

$x$ (radians) shrink hota hai, straight chord length aur arc length ka limiting ratio kya hai?

Forecast: chord arc ke across ek shortcut hai, to chord $<$ arc hamesha. Lekin jab angle shrink hota hai, kya ratio $0$ , kuch fraction, ya $1$ par jaata hai?

Step 1. Picture par lengths set up karo.

Unit circle par arc length $= x$ (yahi reason hai ki Radian Measure use hota hai — angle hi arc hai).
Dono endpoints ke beech chord length $= 2 sin \frac{x}{2}$ (perpendicular daalo; har half $sin \frac{x}{2}$ hai). Yeh step kyun? Hum ek geometry question ko do familiar quantities ke ratio mein convert karte hain.

Step 2. Ratio form karo aur master limit ki shape mein reshape karo: $\frac{chord}{arc} = \frac{2 s i n \frac{x}{2}}{x} = \frac{s i n \frac{x}{2}}{\frac{x}{2}} .$ Yeh step kyun? $x = 2 \cdot \frac{x}{2}$ likhne se sin ka argument denominator ke barabar ho jaata hai — exactly $\frac{s i n u}{u}$ ki shape jahan $u = \frac{x}{2}$ hai.

Step 3. Jab $x \to 0$ , $u = \frac{x}{2} \to 0$ , to ratio $\to 1$ .

Verify: $x = 0.1$ rad: chord $= 2 sin (0.05) = 0.099979$ , arc $= 0.1$ , ratio $= 0.99979 \to 1$ . ✓ "Thin pizza slice" intuition numerically confirm ho gaya.

Example E — degrees, radians nahi (cell 5)

Worked example Compute karo

$x \to 0 lim \frac{sin ( x ^{\circ} )}{x}$ , jahan $x$ degrees mein measure ho.

Forecast: zyaadatar log "1" bol dete hain. Yeh 1 nahi hai. Guess karo kya break hota hai.

Step 1. Degrees ko radians mein convert karo sine ke andar, kyunki $sin$ hamare saari theory mein radians leta hai: $x^{\circ} = \frac{π}{180} x$ radians. Yeh step kyun? Master limit $\frac{s i n u}{u} = 1$ sector area $\frac{1}{2} r^{2} θ$ se prove hua tha, jo sirf radians mein valid hai. To sin ko radian argument dena zaroori hai.

Step 2. Substitute karo aur arguments match karo: $\frac{s i n ( \frac{π}{180} x )}{x} = \frac{s i n ( \frac{π}{180} x )}{\frac{π}{180} x} \cdot \frac{π}{180} .$ Yeh step kyun? Sine ko uska matching denominator $\frac{π}{180} x$ do; correction factor $\frac{π}{180}$ baacha rehta hai.

Step 3. Matched fraction $\to 1$ , to $lim_{x \to 0} \frac{s i n ( x ^{\circ} )}{x} = \frac{π}{180} \approx 0.01745.$

Verify: $x = 1$ degree: $sin (1^{\circ}) = 0.0174524$ , $1$ se divide karo $= 0.0174524 \approx \frac{π}{180}$ . ✓ Saaf " $= 1$ " sirf radians ka privilege tha.

Example F — $e$ limit negative constant ke saath (cell 6)

Worked example Compute karo

$x \to \infty lim (1 - \frac{4}{x})^{x}$ .

Forecast: yeh $1^{\infty}$ hai. Base $1$ se neeche hai, to answer $1$ se neeche hona chahiye — ek decaying quantity. Guess karo kya yeh $e^{4}$ hai ya $e^{- 4}$ .

Step 1. Master $(1 + \frac{a}{x})^{x} \to e^{a}$ se match karo. Yahan $- \frac{4}{x} = \frac{a}{x}$ deta hai $a = - 4$ . Yeh step kyun? Rule $a$ ke sign ki parwah nahi karta; woh sirf $\frac{1}{x}$ ke upar constant padhta hai.

Step 2. (Safety ke liye log check.) $ln (expr) = x ln (1 - \frac{4}{x}) = \frac{l n ( 1 - \frac{4}{x} )}{1/ x}$ . Small $t = - \frac{4}{x}$ ke liye $ln (1 + t) \approx t$ use karo, yeh hai $\frac{- 4/ x}{1/ x} = - 4$ . Yeh step kyun? Log karna $1^{\infty}$ ko $\frac{0}{0}$ mein convert karta hai jise hum evaluate kar sakte hain — indeterminate powers ke liye safe universal method, dekho Indeterminate Forms.

Step 3. $ln (expr) \to - 4$ , to expr $\to e^{- 4} \approx 0.0183$ .

Verify: $x = 100000$ : $(1 - 4/100000)^{100000} \approx 0.018314 \approx e^{- 4} = 0.018316$ . ✓ $1$ se neeche, jaise forecast tha.

Example G — degenerate $e$ : base bahut fast shrink karta hai (cell 7)

Example H — real-world continuous compounding (cell 8)

Worked example Ek bank

6% annual interest offer karta hai, continuously compound hoti hai (infinitely often). Ek deposit 1 saal mein kis factor se grow karti hai? 10 saal mein?

Forecast: 6% simple $\times 1.06$ deta ek saal mein. Continuous compounding thoda zyaada deta hai. Guess karo kitna zyaada.

Step 1. Saal ko $n$ tiny periods mein split karo; har period $\frac{0.06}{n}$ pay karta hai, $n$ times apply hota hai: $factor_{1} = (1 + \frac{0.06}{n})^{n} .$ Yeh step kyun? " $n$ times compounded" ka matlab literally hai base $1 + \frac{rate}{n}$ ko khud se $n$ baar multiply karna.

Step 2. $n \to \infty$ jaane do (continuous). Yeh master hai $a = 0.06$ ke saath: $(1 + \frac{0.06}{n})^{n} \to e^{0.06} \approx 1.06184.$ Yeh step kyun? Continuous = infinitely many splits = exact limit jo $e^{a}$ define karta hai; dekho The Number e and ln.

Step 3. $10$ saal mein exponent scale hota hai: factor $= e^{0.06 \times 10} = e^{0.6} \approx 1.82212$ . Yeh step kyun? Continuous growth ke das saal rate stack karte hain: $e^{r t}$ jahan $r = 0.06$ , $t = 10$ .

Verify: $e^{0.06} = 1.061837$ (vs simple $1.06$ — lagbhag $0.18%$ zyaada, jaise forecast tha). $e^{0.6} = 1.822119$ , to $1000 ki deposit lagbhag $1822.12 ban jaati hai. ✓

Example I — boss fight: dono limits ek saath (cell 9)

Worked example Compute karo

$x \to \infty lim (cos \frac{1}{x})^{x^{2}}$ .

Forecast: jab $x \to \infty$ , $\frac{1}{x} \to 0$ to $cos \frac{1}{x} \to 1$ — phir $1^{\infty}$ . Yahan cosine limit aur $e$ machinery dono chupi hain. Guess karo: kya yeh $1$ , $e^{- 1/2}$ , ya $0$ jaata hai?

Step 1. Log karo: $ln (expr) = x^{2} ln (cos \frac{1}{x})$ . Yeh step kyun? $1^{\infty}$ — pehle log karo, hamesha.

Step 2. $t = \frac{1}{x} \to 0$ set karo. To $x^{2} = \frac{1}{t ^{2}}$ aur humein $\frac{l n ( c o s t )}{t ^{2}}$ chahiye. $ln (cos t) = ln (1 - (1 - cos t)) \approx - (1 - cos t)$ likho small $t$ ke liye: $\frac{l n ( c o s t )}{t ^{2}} \approx \frac{- ( 1 - c o s t )}{t ^{2}} = - \frac{1 - c o s t}{t ^{2}} \to - \frac{1}{2} .$ Yeh step kyun? Yahan sin-family helper $\frac{1 - c o s t}{t ^{2}} \to \frac{1}{2}$ (parent se) log expansion $ln (1 + u) \approx u$ mein seedha plug hota hai. Do atomic limits ek line mein mil rahe hain.

Step 3. $ln (expr) \to - \frac{1}{2}$ , to expr $\to e^{- 1/2} = \frac{1}{e} \approx 0.6065$ .

Verify: $x = 1000$ : $(cos 0.001)^{1 0^{6}} = exp (1 0^{6} ln cos 0.001) = exp (1 0^{6} \cdot (- 5.0 \times 1 0^{- 7})) = exp (- 0.5) \approx 0.60653$ . ✓ $1$ nahi — compounding jeet jaata hai.

Recall

Recall Kaunsa cell kaunsa hai — cover karo aur trap ka naam lo

Example E ki trap ::: degrees mein argument hone par radian value $1$ use karna; sahi answer $\frac{π}{180}$ hai.
Example G ki trap ::: $e$ se pattern-match karna jab base bump $\frac{1}{n ^{2}}$ ho, $\frac{1}{n}$ nahi; answer $1$ hai.
Example F ki trap ::: $a$ ka sign drop karna; answer $e^{- 4}$ hai, $e^{4}$ nahi.
Tool jo Example B ko possible banata hai ::: $sin x$ ki Taylor Series, kyunki $\frac{s i n x}{x}$ akela $x^{3}$ order limit tak nahi pahunch sakta.
Example C ki limit even kyun hai ::: $sin$ odd hai, denominator bhi flip hota hai, minus signs cancel ho jaate hain.

Connections

Squeeze (Sandwich) Theorem — A, C, D ke peeche $\frac{s i n u}{u}$ engine prove karta hai.
Radian Measure — Examples D aur E ka pivot.
Derivative of sin and cos — Example B ka slope same object hai jo Taylor tail ke roop mein dikha.
The Number e and ln — F, G, H mein growth define karta hai.
Indeterminate Forms — har "pehle log karo" moment.
L'Hôpital's Rule — B, F, I ke liye alternative route (lekin atoms ke liye circular hai).
Taylor Series — B aur I ke liye sabse sharp tool.