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📘 CHAPTER: LIMITS

EXERCISE – LIMITS (Revised & Expanded)

Class 11 • Class 12 • IIT-JEE Main & Advanced

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🔹 SECTION A: Limits at Infinity (Basic → Moderate)

Q1. $$ \lim_{x\to\infty} x^{1/x} $$

Q2. $$ \lim_{x\to\infty} \left(x-\sqrt{x^2+x}\right) $$

Q3. $$ \lim_{x\to\infty} \left(1+\frac{1}{\sqrt{x}}\right)^x $$

Q4. $$ \lim_{x\to\infty} \sin(x^2) $$

Q5. $$ \lim_{x\to\infty} \frac{2x^2+3x}{5x^2-x} $$

Q6. $$ \lim_{x\to\infty} \frac{x^3-4}{2x^3+7} $$

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🔹 SECTION B: Exponential & Logarithmic Limits

Q7. $$ \lim_{x\to\infty} \left(\frac{x}{x+2}\right)^x $$

Q8. $$ \lim_{x\to\infty} \frac{x}{\sqrt{3x^2+1}} $$

Q9. $$ \lim_{x\to0} \frac{e^x-1}{x} $$

Q10. $$ \lim_{x\to0} \frac{\ln(1+x)}{x} $$

Q11. $$ \lim_{x\to0} \frac{a^x-1}{x}, \quad (a>0) $$

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🔹 SECTION C: Rational Functions (Dominant Term Method)

Q12. $$ \lim_{x\to\infty} \frac{x}{1+4x^2} $$

Q13. $$ \lim_{x\to\infty} \frac{x^3+2}{x+1} $$

Q14. $$ \lim_{x\to\infty} \frac{6x+1}{2x+5} $$

Q15. $$ \lim_{x\to\infty} \frac{x^2}{1-x^2} $$

Q16. $$ \lim_{x\to\infty} \frac{x^2+4x+6}{3x^2+1} $$

Q17. $$ \lim_{x\to\infty} \frac{1-4x^3}{x^2+2x+1} $$

Q18. $$ \lim_{x\to\infty} \frac{x^2+3}{2x} $$

Q19. $$ \lim_{x\to\infty} \frac{x^4+3x}{x^5+1} $$

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🔹 SECTION D: Limits at a Point (Algebraic & Radical Forms)

Q20. $$ \lim_{x\to0} \frac{\sqrt{x+25}-5}{x} $$

Q21. $$ \lim_{x\to0} \frac{4x-1}{8x-1} $$

Q22. $$ \lim_{x\to4} \frac{\sqrt{x}-2}{x-4} $$

Q23. $$ \lim_{x\to1} \frac{x^5-1}{x-1} $$

Q24. $$ \lim_{x\to0} \frac{\sqrt{1+x}-1}{x} $$

Q25. $$ \lim_{x\to0} \frac{\sqrt{4+x}-2}{x} $$

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🔹 SECTION E: Simplification & Factorisation

Q26. $$ \lim_{x\to0} \frac{(x+5)^2-25}{x} $$

Q27. $$ \lim_{x\to0} \frac{(x+1)^3-1}{x} $$

Q28. $$ \lim_{x\to3} \frac{x^2-9}{x-3} $$

Q29. $$ \lim_{x\to2} \frac{x^2-7x+10}{x-2} $$

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🔹 SECTION F: Trigonometric Limits (JEE Focus)

Q30. $$ \lim_{x\to0} \frac{\sin x}{x} $$

Q31. $$ \lim_{x\to0} \frac{\tan x}{x} $$

Q32. $$ \lim_{x\to0} \frac{1-\cos x}{x^2} $$

Q33. $$ \lim_{x\to0} \frac{\sin 3x}{\sin 5x} $$

Q34. $$ \lim_{x\to0} \frac{\tan 2x}{\sin 3x} $$

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🔹 SECTION G: Advanced & JEE-Level Limits

Q35. $$ \lim_{x\to0} \frac{\sin x-x}{x^3} $$

Q36. $$ \lim_{x\to0} \frac{1-\cos x}{x\sin x} $$

Q37. $$ \lim_{x\to0} \frac{\tan x-\sin x}{x^3} $$

Q38. $$ \lim_{x\to0} \frac{e^x-1-x}{x^2} $$

Q39. $$ \lim_{x\to0} \frac{\ln(1+x)-\sin x}{x^3} $$

Q40. $$ \lim_{x\to\infty} (\sqrt{x^2+x}-x) $$

Q41. $$ \lim_{x\to0} \frac{x-\sin x}{x^3} $$

Q42. $$ \lim_{x\to0} \frac{\tan x-x}{x^3} $$

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📚 Class 9 Science – Complete Table of Contents (विषय-सूची)

CBSE | NCERT Curriculum | Bilingual (English + Hindi)
By Grandmaster Bikram Sutradhar | SirBikramSutradhar | BASTRONAUTWAY

🔬 PHYSICS (भौतिक विज्ञान)

🔢	Chapter Title (English)	अध्याय का नाम (हिंदी)	Page No.
1️⃣	Motion	गति
2️⃣	Force and Laws of Motion	बल तथा गति के नियम
3️⃣	Gravitation	गुरुत्वाकर्षण
4️⃣	Work and Energy	कार्य तथा ऊर्जा
5️⃣	Sound	ध्वनि

🧪 CHEMISTRY (रसायन विज्ञान)

🔢	Chapter Title (English)	अध्याय का नाम (हिंदी)	Page No.
1️⃣	Matter in Our Surroundings	हमारे आस-पास के पदार्थ
2️⃣	Is Matter Around Us Pure?	क्या हमारे आस-पास के पदार्थ शुद्ध हैं?
3️⃣	Atoms and Molecules	परमाणु एवं अणु
4️⃣	Structure of the Atom	परमाणु की संरचना

🧬 BIOLOGY (जीव विज्ञान)

🔢	Chapter Title (English)	अध्याय का नाम (हिंदी)	Page No.
1️⃣	The Fundamental Unit of Life	जीवन की मौलिक इकाई
2️⃣	Tissues	ऊतक
3️⃣	Improvement in Food Resources	धनों में सुधार

📘 ADDITIONAL SECTIONS (अतिरिक्त अनुभाग)