Infinite limits and limits at infinity — vertical - horizontal asymptotes

4.1.4 · Maths › Calculus I — Limits & Derivatives

1. Infinite limits (vertical asymptotes)

KYA hota hai & KYU: Yeh tab hota hai jab tum kisi nonzero number ko $0$ ki taraf jaati cheez se divide karo. Jaise denominator $\to 0$ hota hai, fraction ka size $\to \infty$ ho jaata hai. Result ki sign numerator aur denominator ki signs par depend karti hai $a$ ke paas — isliye hum left aur right alag alag check karte hain.

2. Limits at infinity (horizontal asymptotes)

Rational functions — rule derive karo

Maano $f (x) = \frac{a _{m} x ^{m} + \dots}{b _{n} x ^{n} + \dots}$ . KAISE hum $x \to \infty$ evaluate karte hain: top aur bottom dono ko denominator ki highest power, $x^{n}$ , se divide karo, toh har term ek constant times $1/ x^{k}$ ban jaati hai jise hum $0$ bhej sakte hain.

$\frac{a _{m} x ^{m} + \dots + a _{0}}{b _{n} x ^{n} + \dots + b _{0}} = \frac{a _{m} x ^{m - n} + \dots}{b _{n} + \dots} .$

Agar $m < n$ : numerator mein sirf negative powers hain $\to 0$ . Limit $= 0$ , asymptote $y = 0$ .
Agar $m = n$ : leading terms survive karte hain. Limit $= \frac{a _{m}}{b _{n}}$ , asymptote $y = a_{m} / b_{n}$ .
Agar $m > n$ : numerator bina bound ke grow karta hai. Limit $= \pm \infty$ , koi horizontal asymptote nahi (shayad slant wala ho).

Figure — Infinite limits and limits at infinity — vertical - horizontal asymptotes

3. Forecast-then-Verify drill

Answer padhne se pehle predict karo:

$lim_{x \to \infty} \frac{2 x ^{3} + 1}{x ^{2}}$ ? → degree $3 > 2$ , $+ \infty$ , koi HA nahi.
$lim_{x \to 2^{-}} \frac{1}{x - 2}$ ? → $x - 2 \to 0^{-}$ , toh $\frac{1}{0 ^{-}} = - \infty$ .
$lim_{x \to \infty} e^{- x}$ ? → $e^{- x} = 1/ e^{x} \to 0$ , HA $y = 0$ .
$lim_{x \to \infty} arctan x$ ? → $π /2$ , HA $y = π /2$ (aur $- \infty$ par $- π /2$ ).

Recall Feynman: ek 12-saal ke bacche ko samjhao

Ek road (graph) imagine karo. Ek vertical wall ( $x = a$ ) woh jagah hai jahan road achanak sky mein rocket ki tarah upar jaati hai ya neeche gir jaati hai — tum us spot ke left/right super close ja sakte ho lekin road ki height pagal ho jaati hai. Ek horizontal ceiling ( $y = L$ ) tab hoti hai jab tum forever right chalte ho aur road flat ho jaati hai, ek flat line ke saath chipakti hai lekin kabhi theek use touch nahi karti. Vertical wall = $x$ frozen, height explode. Horizontal ceiling = $x$ bhaagta hai, height shant ho jaati hai.

Active recall #flashcards/maths

Definition of

lim_{x \to a} f (x) = + \infty

in words

Kisi bhi

M

ke liye, ek

δ

hota hai jisse

0 < ∣ x - a ∣ < δ

force karta hai

f (x) > M

x = a

vertical asymptote kab hoti hai?

Jab

a

par koi one-sided limit

+ \infty

- \infty

ho.

Denominator

\to 0

AUR numerator

\to 0

— asymptote hai ya nahi?

Possibly removable hole; pehle factor/cancel karo.

lim_{x \to \infty} 1/ x^{p}

for

p > 0

0

Rational function ki HA jab deg(top)=deg(bottom)?

y =

leading coefficients ka ratio.

HA jab deg(top)>deg(bottom)?

Koi nahi (limit

\pm \infty

hai).

x^{2}

equal hai?

∣ x ∣

; equals

- x

when

x < 0

lim_{x \to - \infty} arctan x

- π /2

lim_{x \to 0^{-}} 1/ x

- \infty

Kisi function mein horizontal asymptotes ki maximum kitni ho sakti hain?

Do (ek

+ \infty

par, ek

- \infty

par).

Connections

Limits — formal epsilon-delta definition
One-sided limits
Continuity and removable discontinuities
Curve sketching using first and second derivatives
Slant (oblique) asymptotes via polynomial division
L'Hôpital's rule for indeterminate forms