Foundations — Limit laws — sum, product, quotient, constant multiple

4.1.2 · D1 · Maths › Calculus I — Limits & Derivatives › Limit laws — sum, product, quotient, constant multiple

Limit laws pe trust karne se pehle, tumhe page par har ek mark mein fluency honi chahiye. Yeh note parent mein use hone wale har ek symbol ko — ek ek karke, bilkul zero se — leta hai, usse ek picture ke saath pair karta hai, aur kehta hai kyun yeh topic uske bina nahi chal sakta. Upar se neeche padho: har item uske upar wale par lean karta hai.

1. The number line aur ek point $a$

Symbol $a$ (ya $2$, ya $3$) woh target hai jisski taraf hum chal rahe hain. Yeh ek fixed, na hilne wala dot hai. Topic ko yeh isliye chahiye kyunki har limit question "as $x$ approaches..." se shuru hota hai — approach karta hai kya? — point $a$ ko.

2. Chaar arithmetic operations $+,-,\times ,÷$

Kyunki poora topic yeh hai ki limits arithmetic mein se "flow" karti hain, hume pehle clearly samajhna hoga ki yeh chaar operations hain kya aur number line par har ek kaisi dikhti hai.

Topic ko yeh kyun chahiye: limit laws ka naam literally Sum, Constant multiple, Product, Quotient hai — har operation ke liye ek law. Yeh kehne ke liye ki "limit $\times$ se guzarti hai", tumhe pehle se jaanna chahiye ki $\times$ ka matlab hai "scale/repeat". Division mein famous fine print ($M\ne 0$) exactly isliye aata hai kyunki tum zero equal parts mein baant nahi sakte.

3. Variable $x$ aur "approaches" ($\to$)

"Approaches" kyun, "equals" kyun nahi? Kyunki limit journey ke baare mein puchti hai, destination ke baare mein nahi. Parent ke Example 3 ki poori power — $\frac{x-3}{x-3}$ ko $x=3$ ke paas cancel karna — tab hi kaam karti hai jab raaste mein $x\ne 3$ ho. Agar arrow ka matlab "equals" hota, toh woh cancellation illegal hoti. "$x$ near $a$" aur "$x=a$" ke beech ka gap hi woh poori wajah hai jiske liye limits exist karti hain.

4. Ek function $f$ aur uska output $f(x)$

Ek graph socho: horizontal axis input road hai, vertical axis output height hai. Floor par har input $x$ ke upar height $f(x)$ par ek single point baitha hai; in points ko jod do toh curve banta hai.

Topic ko yeh kyun chahiye: limit laws saari outputs combine karne ke baare mein hain. Jab hum $f+g$ likhte hain, matlab hai "dono machines ko same $x$ par chalao aur unke outputs add karo" — Section 2 ka addition use karke. Outputs ko name kiye bina combine nahi kar sakte, isliye $f(x)$ aur $g(x)$ pehle aate hain.

5. Limit symbol $\underset{x\to a}{\lim }f(x)=L$

Figure dekho: jab input walker $a$ ki taraf slide karta hai (dono taraf se), curve par ka point ek height $L$ ki taraf slide karta hai. Woh height limit hai.

Figure mein $x=a$ par curve ka open circle dekho — yeh mark karta hai ki hume $a$ par kya hota hai se matlab nahi, sirf curve aas paas kya aim kar raha hai se.

Hume naam $L$ kyun chahiye: laws $L$ aur $M$ ke baare mein statements hain. Yeh kehne ke liye ki "sum $L+M$ ki taraf head karta hai", hume pehle symbols $L$ aur $M$ chaahiye jo "har piece kahan ja raha hai" stand karte hon.

6. Absolute value $|\cdot |$ — "distance" ka symbol

Topic ko yeh kyun chahiye: "close to" ko ek number banna chahiye jise hum shrink kar sakein. Parent ke proofs mein, $|f(x)-L|$ matlab hai "output apne target $L$ se kitna door hai", aur $|x-a|$ matlab hai "input $a$ se kitna door hai". Poora game yeh hai: $|x-a|$ chhota rakho, aur $|f(x)-L|$ chhota rahega. Bina distance symbol ke tum "chhota" nahi keh sakte.

Yeh woh single tool hai jis par parent ke Sum-law proof ka daromdaar hai — yeh hume combined error $|(f-L)+(g-M)|$ ko do alag errors $|f-L|+|g-M|$ se bound karne deta hai, jinmein se har ek ko hum already control karna jaante hain.

7. $\epsilon$ aur $\delta$ — do twin tolerances

${\lim }_{x\to a}f(x)=L$ ke peeche poora sentence yeh hai:

$0<|x-a|<\delta \Rightarrow |f(x)-L|<\epsilon .$

Un symbols ke saath piece by piece padho jo ab tumhare paas hain:

• $|x-a|<\delta$: input $a$ ki distance $\delta$ ke andar hai.
• $0<|x-a|$: lekin $a$ ke equal nahi (woh "approaches, not equals" rule phir se).
• $\Rightarrow$: "force karta hai" / "guarantee karta hai".
• $|f(x)-L|<\epsilon$: toh output $L$ ki distance $\epsilon$ ke andar hai.

Topic ko yeh kyun chahiye: yeh "heading toward" jaise vague word ko ek challenge game mein badal deta hai jiske baare mein hum actually cheezein prove kar sakte hain. Yahi precise definition hai jise parent ke Sum, Product, aur Quotient proofs manipulate karte hain. Poori detail Epsilon-Delta definition of a limit mein hai.

8. Constants $c$, $n$, aur grouping symbols

Atomic limits se pehle, plain bookkeeping marks ka ek last batch:

Yeh bookkeeping marks hain, lekin reader ko proof ke beech mein inpar stumble nahi karna chahiye — isliye inhe yahan ek baar naam de diya gaya hai.

9. Do atomic limits

Parent jo kuch bhi build karta hai woh do itne simple facts par tika hai ki unhe koi laws nahi chahiye:

Yeh kyun matter karte hain: laws sirf un limits ko combine karte hain jo tumhe already pata hain. Yeh do "already known" seeds hain. Inhe laws ke saath add aur multiply karo aur tum parent ke har polynomial aur rational limit tak pahunch sakte ho.

Prerequisite map

Neeche diya diagram (Mermaid mein, ek plain text-to-flowchart syntax — har --> ek arrow hai "feeds into") dikhata hai ki foundations kaise stack up hote hain. Shabdon mein: number line hume point $a$ aur chaar operations deta hai; $a$ se hum "$x$ approaches $a$" build karte hain, phir functions aur unke outputs, phir limit $L$. Alag se, absolute value aur triangle inequality $\epsilon$–$\delta$ game ko feed karti hain. Limit idea aur $\epsilon$–$\delta$ game milke do atomic limits dete hain, aur sab kuch milke limit laws power karta hai.

Equipment checklist

Har line ka format Question ::: Answer hai. ::: ek reveal separator hai — iske baad sab kuch cover karo, jawaab dene ki koshish karo, phir hidden text se check karo.

Connections

• Epsilon-Delta definition of a limit — jahan Section 7 ka $\epsilon$–$\delta$ game poori tarah develop kiya gaya hai.
• One-sided limits — Section 3 ki "baayein se / daayein se" walk, alag alag split ki gayi.
• Continuity — woh case jahan $L$ actually $f(a)$ ke equal hota hai.
• Parent: Limit laws — sum, product, quotient, constant multiple.