📘 CHAPTER: LIMITS

EXERCISE – LIMITS (Revised & Expanded)

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

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

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

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

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

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

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

🔹 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)

🧪 CHEMISTRY (रसायन विज्ञान)

🧬 BIOLOGY (जीव विज्ञान)

📘 ADDITIONAL SECTIONS (अतिरिक्त अनुभाग)