4.1.6 · D1Important limits — lim(sin x - x) = 1, lim((1+1 - n)ⁿ) = e
Dono famous limits ek hi chhota sa sawaal poochh rahe hain: "jab ek input zero ki taraf shrink karta hai (ya ek count infinity ki taraf badhta hai), toh koi particular ratio ya product kis fixed number par ja ke theharta hai?" Poora topic bas yahi seekhna hai ki us 'theharne' ke behaviour ko padhna kaise aata hai — isliye pehle hume har wo symbol fluently samajhni chahiye jo use karte hain usse likhne ke liye: angle, radian, sine, ratio, limit arrow, aur exponent.
Neeche har wo notation hai jis par parent note depend karta hai, is tarah order ki gayi hai ki har ek cheez pichli cheez se bani ho. Koi bhi cheez use se pehle define ki jaati hai.
→ x → 0 x approaches 0 nahin hai "x 0 x 0 0 n → ∞ n 1 , 2 , 3 , …
Definition Two-sided vs one-sided —0 dono directions se approach karo
Zero ke do neighbourhoods hain: usse thode upar ke numbers (+ 0.1 , + 0.01 , … x → 0 + − 0.1 , − 0.01 , … x → 0 − x → 0 dono sides se same number par settle ho. Agar dono sides alag-alag results dete, toh two-sided limit exist hi nahin karta.
Figure dekho. Top line par burnt-orange dots 0 right se creep kar rahe hain (x → 0 + aur plum dots 0 left se creep kar rahe hain (x → 0 − teal dots bade se bade n → motion hai, koi destination nahin jahan tum pohonchte ho.
x s i n x x s i n x even function hai: x − x sin ( − x ) = − sin x − x − s i n x = x s i n x x → 0 + x → 0 − identically hota hai — dono sides automatically match karti hain, aur two-sided limit ke baare mein baat karna safe hai. (n → ∞ n
Intuition Arrow ki zaroorat kyun haiAgar hum seedha x = 0 x s i n x 0 0 0 sneak up karte hain aur dekhte hain value kahan ja rahi hai. Wahi sneaking-up exactly hai jo →
lim lim x → 0 ( some expression in x ) (koi expression) settle karta hai jab x 0 dono sides se approach karta hai". lim ke neeche likha chhota $x\to 0$ batata hai kis taraf input move kar raha hai — aur bina + − dono taraf ek saath .
Worked example Isko zaor se padhnax → 0 lim x sin x = 1 "jab x x s i n x 1 1
Yeh poore chapter mein sabse important single symbol hai, aur parent topic isko har line mein use karta hai.
Definition Radian — angle naapne ka honest tarikaRadius 1 unit circle , toh uska radius r = 1 ( 1 , 0 ) curved edge ke saath saath chalo . Kisi angle ka radian measure simply hai tum us edge par kitna chale . Distance 1 1 2 π 2 π
Intuition Figure mein KYA observe karna haiOrange thick arc woh distance hai jo tum chale; teal radius batata hai tum kahan ruke. Dono labels agree karte hain: arc length = 1 aur angle = 1 picture par literally same number hain. Yahi equality radians deta hai — aur yeh sirf radians mein sach hai. Degrees mein, distance 1 ≈ 57.3° x s i n x → 1 x Radian Measure dekho.
sin x x Horizontal axis tak ek seedhi line neeche girao. Us vertical drop ki length sin x sin x height hai.
Intuition Figure mein KYA observe karna haiEk chhote angle x
orange arc (length = x teal chord (seedha shortcut across),plum vertical (length = sin x
Plum segment follow karo: yeh height hai, aur yeh apne saath wale orange arc se zara hi chhoti hai. Jab x = x = sin x same length hote hain, toh unka ratio x s i n x 1 Squeeze (Sandwich) Theorem ka kaam hai.
cos x tan x
cos x horizontal position (kitna right hai).tan x = cos x sin x angle ka edge , extend hone par, ( 1 , 0 )
tan x chhote triangle (height sin x bade triangle (height tan x cos x < 1 tan x = c o s x s i n x bada hota hai sin x Derivative of sin and cos is same limit se seedha bana hai.
b a b a "b a Ek ratio ki limit, jaise x s i n x jab upar aur neeche dono shrink karte hain, kya woh step mein rehte hain? Agar upar aur neeche same speed se shrink karein, toh ratio 1 0
e e e ≈ 2.71828 nahin karte; hum ise exactly us value ke roop mein define karte hain jis par compounding expression settle karta hai:
e := lim n → ∞ ( 1 + n 1 ) n . π e The Number e and ln mein hai.
Definition "Indeterminate" = notation akela decide nahin kar sakta
0 0 0 nahin padh sakte — yeh 0 1 7 ∞ speeds ke hisaab se. x s i n x x → 0 dono sides se, jo agree karte hain kyunki function even hai, §1) ek 0 0 1 1 ∞ 1 ∞ n → ∞ 1 + n 1 n 1 1 + n 1 compound karta hai. ( 1 + n 1 ) n e
Intuition Figure mein KYA observe karna haiDo panels. Left: curve x s i n x teal line at 1 par flatten hota hai jab x → 0 dono sides se — naive guess 0 Right: dots ( 1 + n 1 ) n orange line at e ≈ 2.718 par ruk jaate hain — naive guess 1 wajah hai ki in limits ka poora ek topic hai. Zyada cases Indeterminate Forms mein hain.
( 1 + n 1 ) n ( 1 + n 1 ) n number ( 1 + n 1 ) n Do cheezein ladte hain: base 1 + n 1 n 1 = 1 n e
Intuition Bank ki picture (e n n 1 n → ∞ e ≈ 2.718 The Number e and ln mein develop kiya gaya.
ln e ( ) ln ln ( e a ) = a ln ( A b ) = b ln A exponent ko neeche ek ordinary product mein kheench laata hai . Exactly aise hi parent note ( 1 + n 1 ) n ln n 0 0 L'Hôpital's Rule , aur exact-polynomial view hai Taylor Series .
Approach arrow x to 0 both sides
Radian measure r equals 1
Number e defined as a limit
Indeterminate 1 to infinity
"x → 0 "x 0 dono sides se kareebi aata jaata hai lekin kabhi us par pohonchta nahin."
Plain (unsigned) lim x → 0 Right (0 + 0 −
x s i n x Function even hai, isliye left aur right behaviour identical hain.
Unit circle use karke radian define karo Radius-1
In limits ke liye angle radians mein kyun hona chahiye? Sirf radians mein arc length angle ke barabar hoti hai, aur sector area = 2 1 r 2 x 2 1 x r = 1
sin x Unit-circle point ki horizontal axis ke upar vertical height.
Proof ko tan x tan x = c o s x s i n x > sin x
Number e e := lim n → ∞ ( 1 + n 1 ) n ≈ 2.718
Ek indeterminate 0 0 Upar kitni tezi se shrink karta hai bottom ki tulna mein.
1 ∞ 1 Base 1 + n 1
ln A b Ise b ln A
