Question bank — Intuitive concept of a limit — table of values, graphical

True — limit sirf $x$ ko $a$ ke paas inspect karti hai, kabhi $x=a$ ko nahi, isliye $a$ par hole hone se approached value untouched rehti hai.

False — point par value jaanna tumhe approach ke baare mein kuch nahi batata; ek stray dot $L$ par baith sakta hai jabki curve kahin aur ja rahi ho. (Dono ka agree karna exactly Continuity at a point hai.)

False — woh equal bhi hone chahiye; alag-alag exist karna kaafi nahi, jaise ek jump dikhata hai left $=1$, right $=2$, dono exist karte hain, phir bhi disagree karte hain.

Proof ke roop mein False — ek table sirf ek forecast ke liye evidence build karti hai; yeh kabhi trend guarantee nahi kar sakti, kyunki ek function kisi bhi value par veer kar sakta hai jo tumne tabulate ki usse bhi chhoti ho.

False — limit ek endless squeeze ka target hai; outputs $2$ ke paas hamesha ke liye approach kar sakte hain bina kisi $x\ne a$ ke usse chhue.

False — $f(a)$ ek jump par bilkul perfectly defined ho sakta hai; limit isliye fail hoti hai kyunki dono sides disagree karti hain, point missing hai isliye nahi.

False — limit lone point $x=a$ ko bilkul ignore karti hai, isliye us ek value ko edit karne se approach ke baare mein kuch nahi badalta.

True — limit sirf surrounding values padhti hai, jo identical hain, isliye $a$ par disagreement usse invisible hai.

Error yeh hai ki $\frac{0}{0}$ ko final answer treat kiya gaya; yeh ek indeterminate form hai jo pehle simplify karo ka signal deta hai — factor karo aur cancel karo toh $x+1\to 2$ milta hai.

Yeh sirf $x\ne 1$ ke liye agree karte hain; original $1$ par undefined hai jabki $x+1$ nahi. Cancellation limit ke andar legal hai precisely isliye kyunki $x\to 1$ mein $x\ne 1$ rehta hai.

Ek side kaafi nahi hai — right-hand limit kuch bhi ho sakta hai. Two-sided limit naam dene se pehle tumhe dono sides confirm karni hongi aur yeh ki woh agree karte hain.

Substitution sirf ek first guess hai jo nice (continuous) functions ke liye kaam karta hai; holes, jumps, ya $0/0$ traps par yeh undefined ya galat value deta hai.

"Almost $2$" jaisi koi number nahi hoti; limit us single value ke roop mein define hoti hai jiske paas outputs arbitrarily close aate hain, aur woh value exactly $2$ hai.

Yeh simplification sirf $x>0$ ke liye hold karti hai; $x<0$ ke liye yeh $-1$ equals hoti hai. Left limit $-1\ne$ right limit $+1$, isliye two-sided limit exist nahi karta.

Ek hole sirf point $(1,2)$ ko remove karta hai, height $2$ ki taraf dono sides se approach ko nahi — isliye limit $2$ hai hole ke bawajood.

Kyunki poora purpose broken spots par well-behaved approach describe karna hai; $x=a$ allow karne se wahi division-by-zero ya undefined value wapas aa jaati jo hum step around karne ki koshish kar rahe hain.

Ek limit ek single destination naam deta hai; agar dono approach paths alag heights ki taraf jaate hain toh koi ek destination nahi hai, isliye kisi ek ko "limit" kehna arbitrary hoga. (Dekho One-sided limits and limits at infinity.)

Ek table finitely many $x$-values sample karti hai; "arbitrarily close" ki sachchi guarantee ke liye Formal epsilon-delta definition of a limit chahiye, jo har nearness ko ek saath handle karta hai, na sirf woh rows jo tumne likhe.

$0/0$ ek shared factor se aata hai jo top aur bottom dono par vanish hota hai; usse cancel karne se woh finite value expose hoti hai jo ratio actually heading toward tha jab fake cancellation remove hoti hai.

Limit nearby outputs ka trend padhti hai (sabhi $2$ ke paas); $x=1$ par lone value kisi bhi doosre point ke "paas" nahi hai, isliye yeh kabhi us squeeze mein enter nahi karti jo limit define karti hai.

"Undefined" ka matlab koi value nahi; "indeterminate" ka matlab hai ki form akele answer decide nahi karti — alag-alag $0/0$ expressions alag-alag numbers tak limit kar sakte hain, isliye aur kaam zaroori hai.

Kisi curve ka slope ek single point par vanishing rise over vanishing run ka $\frac{0}{0}$ ratio hai; sirf ek limit hi hume woh value naam dene deta hai jis par woh ratio approach karta hai. (Yeh hai The derivative as a limit of difference quotients.)

Nahi — jab $x\to 0$ toh input $\frac{1}{x}$ saari values mein race karta hai, isliye output endlessly $-1$ aur $1$ ke beech oscillate karta hai aur kabhi ek destination par settle nahi hota.

Nahi — "limit $+\infty$ hai" behaviour (unbounded growth) describe karta hai lekin koi finite value approached nahi hoti, isliye wahan ek finite limit exist nahi karta.

Two-sided limit ke roop mein nahi — tum sirf us side se one-sided limit le sakte ho jahan function rehta hai; missing side ke paas approach karne ke liye kuch nahi. (Dekho One-sided limits and limits at infinity.)

Kaafi nahi — do alag constants ke beech bound karna wander karne ki jagah chhodta hai. Tumhe aisi bounds chahiye jo dono same value ki taraf approach karein, jo Squeeze (Sandwich) Theorem hai.

Haan — har side individually apni khud ki height par home in karti hai, isliye dono one-sided limits exist karte hain; two-sided limit sirf isliye fail hoti hai kyunki woh dono heights alag hain.

Haan — har nearby output $0$ hai, isliye approached value $0$ hai chahe $a$ par kuch bhi ho; constant functions apne constant ki taraf har jagah approach karte hain.

Yeh defined nahi hai — koi doosre points approach karne ke liye nahi hain, koi trend read karne ko nahi hai; limit concept simply apply nahi hota.