Limits Approximation Trick in 59 Seconds! 🔥 | JEE Maths

 

❓ Concept

🎬 Absolute Max–Min Trick

JEE mein jab question bole absolute maximum / minimum,
toh sirf derivative nikaalna kaafi nahi hota.

👉 Real game interval checking ka hota hai.

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✍️ Short Explanation

Derivative hume possible candidates deta hai,
lekin final decision hamesha function values se hota hai.

Isliye JEE mein absolute max–min ke liye
ek fixed rule follow kiya jaata hai.

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🔹 Step 1 — Meaning of Critical Points

Critical points woh $x$-values hote hain jahan:

$$f'(x) = 0 \quad \text{or} \quad f'(x) \text{ is undefined}$$

📌 Important:

• Ye sirf candidates hote hain

• Guaranteed max/min nahi hote

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🔹 Step 2 — Closed Interval Rule (MOST IMPORTANT 🔥)

Agar interval ho:

$$[a, b]$$

Toh absolute maximum aur minimum ke liye
sirf aur sirf yeh points check hote hain:

1️⃣ Har critical point jo $(a,b)$ ke andar ho
2️⃣ Endpoint $x = a$
3️⃣ Endpoint $x = b$

Then:

• Largest value → Absolute Maximum

• Smallest value → Absolute Minimum

📌 Is rule ko ignore karna = direct marks loss.

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🔹 Step 3 — log|x| Type Functions

Agar function mein:

$$\log|x|$$

Toh yaad rakho:

• Derivative $x=0$ par undefined hota hai

• Isliye $x=0$ critical point ban jaata hai

⚠️ Lekin:

• Sirf tab consider karo jab $0$ interval ke andar ho

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🔹 Step 4 — Critical Point ≠ Max/Min Always

Har critical point:

• Local maximum ho sakta hai

• Local minimum ho sakta hai

• Ya kuch bhi nahi (inflection point)

👉 Sirf value nikaalne se sach pata chalta hai,
graph ya intuition pe bharosa mat karo.

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🔹 Step 5 — JEE Strategy (Golden Framework ✅)

Har absolute max–min question ke liye ye checklist follow karo:

1️⃣ Function differentiate karo
2️⃣ Critical points nikaalo
3️⃣ Sirf woh points lo jo interval ke andar ho
4️⃣ $f(a)$, $f(b)$, $f(\text{critical})$evaluate karo
5️⃣ Highest & lowest value identify karo
6️⃣ Question ke hisaab se compute karo:

• $M + m$
  

• $|M - m|$
  

• $M/m$, etc.

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✅ Final Takeaway

🧠 Golden Rule:

• Derivative = candidate finder

• Interval checking = decision maker

Agar interval check miss hua,
👉 absolute max–min kabhi correct nahi aayega.