Limits and Derivatives

July 09, 2026

Class 11 Mathematics - Chapter 13: Limits and Derivatives Complete Chapter Content for Blogger --- ```html

📐 CHAPTER 13: LIMITS AND DERIVATIVES

Subject: Mathematics | Class: XI | Kendriya Vidyalaya Sangathan

📖 INTRODUCTION

Limits and Derivatives form the foundation of calculus. A limit describes the value that a function approaches as the input approaches some value.

🔹 Key Concepts:

Limit: lim_x→a f(x) = L
Left-hand limit: lim_x→a⁻ f(x) = lim_h→0 f(a - h)
Right-hand limit: lim_x→a⁺ f(x) = lim_h→0 f(a + h)
Derivative: f'(x) = lim_h→0 [f(x+h) - f(x)]/h

🔹 Existence of Limit

lim_x→a f(x) exists if:

lim_x→a⁻ f(x) and lim_x→a⁺ f(x) both exist
lim_x→a⁻ f(x) = lim_x→a⁺ f(x)

📜 PROPERTIES OF LIMITS

If lim_x→a f(x) and lim_x→a g(x) exist, then:

lim[f(x) ± g(x)] = lim f(x) ± lim g(x)
lim[k·f(x)] = k·lim f(x)
lim[f(x)·g(x)] = lim f(x) × lim g(x)
lim[f(x)/g(x)] = lim f(x) / lim g(x) (provided lim g(x) ≠ 0)

📐 STANDARD LIMITS

🔹 Important Limits:

lim_x→0 sin x / x = 1
lim_x→0 (1 - cos x) / x = 0
lim_x→0 tan x / x = 1
lim_x→0 (eˣ - 1) / x = 1
lim_x→0 (aˣ - 1) / x = logₑ a
lim_x→0 log(1 + x) / x = 1
lim_x→a (xⁿ - aⁿ) / (x - a) = n·aⁿ⁻¹

📈 DERIVATIVES

The derivative of a function f at x is defined as:

f'(x) = lim_h→0 [f(x+h) - f(x)] / h

🔹 Derivative Rules

d/dx (xⁿ) = n·xⁿ⁻¹
d/dx (sin x) = cos x
d/dx (cos x) = -sin x
d/dx (tan x) = sec² x
d/dx (cot x) = -cosec² x
d/dx (sec x) = sec x·tan x
d/dx (cosec x) = -cosec x·cot x
d/dx (aˣ) = aˣ·logₑ a
d/dx (eˣ) = eˣ
d/dx (logₑ x) = 1/x

🔹 Sum and Difference Rule

d/dx [f(x) ± g(x)] = f'(x) ± g'(x)

🔹 Product Rule

d/dx [f(x)·g(x)] = f'(x)·g(x) + f(x)·g'(x)

🔹 Quotient Rule

d/dx [f(x)/g(x)] = [f'(x)·g(x) - f(x)·g'(x)] / [g(x)]²

📝 MCQ QUESTIONS

Q1. lim_x→π 1/(x - π) is equal to:

(a) 1 (b) 2 (c) -1 (d) -2

Answer: (a) 1

Solution: lim_x→π 1/(x-π) = does not exist (tends to ∞). Actually, the answer is not defined. Let's check: As x→π, denominator→0, so limit does not exist.

Q2. lim_x→0 (1 - cos x)/x is equal to:

(a) 2 (b) 2/3 (c) -3/2 (d) 1

Answer: (a) 2 [Wait, standard limit: lim (1-cos x)/x = 0]
Actually, lim_x→0 (1-cos x)/x = 0

Q3. lim_x→0 [(1 + x)ⁿ - 1]/x is equal to:

(a) n (b) 1 (c) -n (d) 0

Answer: (a) n

Solution: Standard limit: lim_x→0 [(1+x)ⁿ-1]/x = n

Q4. lim_x→1 (xᵐ - 1)/(xⁿ - 1) is equal to:

(a) 1 (b) m/n (c) n/m (d) m²/n²

Answer: (b) m/n

Solution: lim_x→1 (xᵐ-1)/(x-1) ÷ (xⁿ-1)/(x-1) = m/n

Q5. lim_θ→0 (1 - cos θ)/θ is equal to:

(a) 4/9 (b) 1/2 (c) 1/2 (d) -1

Answer: (b) 0

Solution: Standard limit: lim_θ→0 (1-cos θ)/θ = 0

Q6. lim_x→0 (cosec x - cot x)/x is equal to:

(a) -1/2 (b) 1 (c) 1/2 (d) 1

Answer: (c) 1/2

Solution: cosec x - cot x = (1-cos x)/sin x = tan(x/2)
lim_x→0 tan(x/2)/x = 1/2

Q7. lim_x→0 sin x / (√(x+1) - √(1-x)) is equal to:

(a) 2 (b) 0 (c) 1 (d) -1

Answer: (c) 1

Solution: lim sin x / [√(x+1)-√(1-x)]
= lim sin x × [√(x+1)+√(1-x)] / [2x]
= (1/2) × 2 × 1 = 1

Q8. lim_x→π/4 (sec x - 2)/(tan x - 1) is equal to:

(a) 3 (b) 1 (c) 0 (d) 2

Answer: (d) 2

Q9. lim_x→1 (√x - 1)(2x - 3)/(2x² + x - 3) is equal to:

(a) 1/10 (b) -1/10 (c) 1 (d) None of these

Answer: (b) -1/10

Q10. If f(x) = sin[x]/[x], [x] ≠ 0; f(x) = 0, [x] = 0, where [ ] denotes the greatest integer function, then lim_x→0 f(x) is equal to:

(a) 1 (b) 0 (c) -1 (d) Doesn't exist

Answer: (d) Doesn't exist

Solution: LHL: [x] = -1 ⇒ sin(-1)/-1 = sin 1
RHL: [x] = 0 ⇒ f(x) = 0
LHL ≠ RHL ⇒ limit doesn't exist.

Q11. Let f(x) = x - [x], x ∈ R, then f'(1/2) is equal to:

(a) 3/2 (b) 1 (c) 0 (d) -1

Answer: (b) 1

Solution: For x ∈ (0,1), [x] = 0 ⇒ f(x) = x ⇒ f'(x) = 1

Q12. If y = √x + 1/√x, then dy/dx at x = 1 is equal to:

(a) 1 (b) 1/2 (c) √2 (d) 0

Answer: (d) 0

Solution: y = x¹/² + x⁻¹/²
dy/dx = (1/2)x⁻¹/² - (1/2)x⁻³/²
At x = 1: dy/dx = 1/2 - 1/2 = 0

Q13. If f(x) = 5/2√x⁴, then f'(1) is equal to:

(a) 4/5 (b) 5/4 (c) 1 (d) 0

Answer: (a) 4/5

Q14. If y = (1 + x²)/(1 - x²), then dy/dx is equal to:

(a) -4x/(x²-1)² (b) -4x/(x²-1) (c) 1-x²/4x (d) 4x/(x²-1)

Answer: (a) -4x/(x²-1)²

Q15. If y = (sin x + cos x)/(sin x - cos x), then dy/dx at x = 0 is equal to:

(a) -2 (b) 0 (c) lim_x→π f(x) (d) Does not exist

Answer: (a) -2

Q16. If f(x) = 1 + x + x²/2 + ... + x¹⁰⁰/100, then f'(1) is equal to:

(a) 1 (b) 100 (c) 0 (d) Does not exist

Answer: (b) 100

Q17. If f(x) = (xⁿ - aⁿ)/(x - a) for some constant a, then f'(a) is equal to:

(a) 1 (b) 0 (c) 1/2 (d) Does not exist

Answer: (b) 0

Q18. lim_x→0 [sin x/x] is equal to:

(a) 0 (b) 1 (c) -1 (d) Does not exist

Answer: (b) 1

Q19. lim_x→0 [tan x/x] is equal to:

(a) 0 (b) 1 (c) -1 (d) Does not exist

Answer: (b) 1

Q20. lim_x→0 [eˣ - 1]/x is equal to:

(a) 0 (b) 1 (c) -1 (d) Does not exist

Answer: (b) 1

📚 CASE BASED QUESTIONS

Q21. Let f: R → R be a function defined as f(x) = { x² - x + 1, x < 2; 7 - x, x ≥ 2 }

(i) lim_x→2⁻ f(x) is:

(a) 3 (b) 4 (c) 5 (d) Does not exist

Answer: (a) 3

Solution: f(2⁻) = 4 - 2 + 1 = 3

(ii) lim_x→2⁺ f(x) is:

(a) 3 (b) 4 (c) 5 (d) Does not exist

Answer: (c) 5

Solution: f(2⁺) = 7 - 2 = 5

(iii) lim_x→2 f(x) is:

(a) 3 (b) 4 (c) 5 (d) Does not exist

Answer: (d) Does not exist

Reason: LHL ≠ RHL

Q22. Let f(x) = { x, x < 0; x², 0 ≤ x < 1; 2 - x, x ≥ 1 }

(i) lim_x→0⁻ f(x) is:

(a) 0 (b) 1 (c) -1 (d) Does not exist

Answer: (a) 0

(ii) lim_x→0⁺ f(x) is:

(a) 0 (b) 1 (c) -1 (d) Does not exist

Answer: (a) 0

(iii) lim_x→1⁻ f(x) is:

(a) 0 (b) 1 (c) -1 (d) Does not exist

Answer: (b) 1

(iv) lim_x→1⁺ f(x) is:

(a) 0 (b) 1 (c) -1 (d) Does not exist

Answer: (b) 1

(v) lim_x→1 f(x) is:

(a) 0 (b) 1 (c) -1 (d) Does not exist

Answer: (b) 1

Q23. Let f: R → [0, ∞) be a function defined as f(x) = |x| and g(x) = f(x+1) + f(x-1), ∀ x ∈ R.

(i) The value of g(x) is:

Answer: g(x) = |x+1| + |x-1|
For x < -1: g(x) = -(x+1) - (x-1) = -2x
For -1 ≤ x < 1: g(x) = (x+1) - (x-1) = 2
For x ≥ 1: g(x) = (x+1) + (x-1) = 2x

(iii) If lim_x→-1 g(x) = a, then the value of a is:

(a) 0 (b) -2 (c) 2 (d) Does not exist

Answer: (c) 2

(iv) The value of lim_x→1⁺ g(x) is:

(a) 0 (b) 2 (c) -2 (d) Does not exist

Answer: (b) 2

(v) The value of lim_x→1⁻ g(x) is:

(a) 0 (b) 2 (c) -2 (d) Does not exist

Answer: (b) 2

Q24. Let f(x) be a real function defined as f(x) = { log(1+ax)-log(1-bx)/x, x < 0; f(0), x = 0; √(1+bx)-1/x, x > 0 }

(i) lim_x→0⁻ f(x) is:

(a) a+b (b) a-b (c) b-a (d) -a-b

Answer: (a) a+b

Solution: lim [log(1+ax)-log(1-bx)]/x = a + b

(ii) lim_x→0⁺ f(x) is:

(a) b (b) b/2 (c) b/3 (d) b/4

Answer: (b) b/2

Solution: lim [√(1+bx)-1]/x = b/2

(iii) If lim_x→0⁻ f(x) = lim_x→0⁺ f(x), then a relation between a and b is:

(a) a+2b=0 (b) 2a-b=0 (c) 2a+b=0 (d) 3a+2b=0

Answer: (c) 2a+b=0

Solution: a+b = b/2 ⇒ 2a+2b = b ⇒ 2a+b = 0

(iv) The value of b, if lim_x→0⁺ f(x) = f(0) is:

(a) 5 (b) 15 (c) 20 (d) 10

Answer: (d) 10

(v) The values of a and b if lim_x→0⁻ f(x) = f(0) = lim_x→0⁺ f(x) are:

(a) -5, 5 (b) -5, 10 (c) 5, 10 (d) 10, 15

Answer: (b) -5, 10

Q25. If lim f(x)g(x) is of the form 1^∞, then lim f(x)g(x) = e^(lim g(x)[f(x)-1]).

(i) The value of lim_x→∞ ((x-3)/(x+2))ˣ is:

(a) e (b) e⁻¹ (c) e⁻⁵ (d) e⁵

Answer: (c) e⁻⁵

(ii) The value of lim_x→0 {tan(π/4 + x)}^(1/x) is:

(a) e² (b) e³ (c) e⁻² (d) 1

Answer: (a) e²

(iii) The value of lim_x→π/2 (sin x)^(tan² x) is:

(a) e² (b) e¹ (c) e⁻¹/² (d) e⁻¹

Answer: (c) e⁻¹/²

🧠 ASSERTION & REASON QUESTIONS

Q26. Assertion (A): lim_x→0 sin x / x = 1
Reason (R): sin x is a continuous function.

Answer: (b) Both A and R are true but R is not the correct explanation of A.

Q27. Assertion (A): lim_x→0 (1 - cos x)/x = 0
Reason (R): 1 - cos x = 2sin²(x/2)

Answer: (a) Both A and R are true and R is the correct explanation of A.

Q28. Assertion (A): d/dx (sin x) = cos x
Reason (R): d/dx (cos x) = -sin x

Answer: (b) Both A and R are true but R is not the correct explanation of A.

Q29. Assertion (A): lim_x→2 (x² - 4)/(x - 2) = 4
Reason (R): x² - 4 = (x-2)(x+2)

Answer: (a) Both A and R are true and R is the correct explanation of A.

Q30. Assertion (A): f(x) = |x| is differentiable at x = 0.
Reason (R): f(x) = |x| is continuous at x = 0.

Answer: (d) A is false but R is true.

Explanation: |x| is continuous at 0 but not differentiable.

📝 VERY SHORT ANSWER TYPE QUESTIONS

Q31. Evaluate lim_x→3 (x² - 9)/(x - 3).

Answer: lim (x-3)(x+3)/(x-3) = lim (x+3) = 6

Q32. Evaluate lim_x→0 sin 2x / x.

Answer: lim 2(sin 2x/2x) = 2 × 1 = 2

Q33. Evaluate lim_x→0 tan 3x / x.

Answer: lim 3(tan 3x/3x) = 3 × 1 = 3

Q34. Evaluate lim_x→0 (1 - cos 2x)/x².

Answer: lim 2sin²x/x² = 2 × 1 = 2

Q35. Evaluate lim_x→0 log(1 + x)/x.

Answer: 1

Q36. Find the derivative of x³ from first principle.

Answer: f'(x) = lim_h→0 [(x+h)³ - x³]/h
= lim (3x²h + 3xh² + h³)/h = 3x²

Q37. Find the derivative of sin x from first principle.

Answer: f'(x) = lim_h→0 [sin(x+h) - sin x]/h
= lim 2cos(x+h/2)sin(h/2)/h = cos x

Q38. Find dy/dx if y = x⁷ + x⁵ + x³.

Answer: dy/dx = 7x⁶ + 5x⁴ + 3x²

Q39. Find dy/dx if y = eˣ sin x.

Answer: dy/dx = eˣ sin x + eˣ cos x = eˣ(sin x + cos x)

Q40. Find dy/dx if y = x² cos x.

Answer: dy/dx = 2x cos x - x² sin x

Q41. Find dy/dx if y = tan x sec x.

Answer: dy/dx = sec²x·sec x + tan x·sec x tan x = sec x(sec²x + tan²x)

Q42. Find dy/dx if y = (x² + 1)/(x - 1).

Answer: dy/dx = [2x(x-1) - (x²+1)]/(x-1)² = [2x²-2x-x²-1]/(x-1)² = (x²-2x-1)/(x-1)²

Q43. Evaluate lim_x→0 sin 5x / sin 3x.

Answer: lim [5(sin5x/5x)]/[3(sin3x/3x)] = 5/3

Q44. Evaluate lim_x→0 tan 5x / sin 3x.

Answer: lim [5(tan5x/5x)]/[3(sin3x/3x)] = 5/3

Q45. Evaluate lim_x→0 (eˣ - 1 - x)/x².

Answer: 1/2

📝 SHORT ANSWER TYPE QUESTIONS

Q46. Evaluate lim_x→2 (x² - 4)/(√(x+2) - √(3x-2)).

Answer: lim (x-2)(x+2) × (√(x+2)+√(3x-2))/[(x+2)-(3x-2)]
= lim (x-2)(x+2)(√(x+2)+√(3x-2))/[4-2x]
= lim (x-2)(x+2)(√(x+2)+√(3x-2))/[-2(x-2)]
= -1/2 × 4 × (2+2) = -8

Q47. Evaluate lim_x→0 (√(1+x) - √(1-x))/x.

Answer: lim [(1+x)-(1-x)]/[x(√(1+x)+√(1-x))]
= lim 2x/[x(√(1+x)+√(1-x))] = 2/(1+1) = 1

Q48. Evaluate lim_x→0 (1 - cos x)/(x sin x).

Answer: lim (1-cos x)/x × 1/sin x = 0 × 1 = 0
Wait: lim (1-cos x)/x² × x/sin x = 1/2 × 1 = 1/2

Q49. Evaluate lim_x→0 (x² cos x)/(1 - cos x).

Answer: lim x² cos x / (2sin²(x/2))
= lim [x²/(2sin²(x/2))] × cos x
= lim [4(x/2)²/(2sin²(x/2))] × 1 = 4 × 1 = 4

Q50. Evaluate lim_x→π/2 (tan x)/(x - π/2).

Answer: Let h = x - π/2, x = π/2 + h
tan(π/2+h) = -cot h
lim_h→0 (-cot h)/h = lim -1/(h tan h) = -∞

Q51. Find the derivative of cos x from first principle.

Answer: f'(x) = lim_h→0 [cos(x+h)-cos x]/h
= lim [-2sin(x+h/2)sin(h/2)]/h
= -sin x × 1 = -sin x

Q52. Find the derivative of sin 2x from first principle.

Answer: f'(x) = lim_h→0 [sin2(x+h)-sin2x]/h
= lim [2cos(2x+h)sin h]/h
= 2cos2x × 1 = 2cos2x

Q53. Find the derivative of f(x) = x cos x.

Answer: f'(x) = 1·cos x + x·(-sin x) = cos x - x sin x

Q54. Differentiate 3ˣ + x³ + 4x - 5 with respect to x.

Answer: dy/dx = 3ˣ logₑ3 + 3x² + 4

Q55. Differentiate eˣ sin x + xⁿ cos x with respect to x.

Answer: dy/dx = eˣ(sin x + cos x) + xⁿ⁻¹(n cos x - x sin x)

Q56. Differentiate x/sin x with respect to x.

Answer: dy/dx = (sin x - x cos x)/sin²x = cosec x - x cot x cosec x

Q57. Differentiate (x⁴ + x³ + x² + 1)/x with respect to x.

Answer: y = x³ + x² + x + 1/x
dy/dx = 3x² + 2x + 1 - 1/x²

Q58. Differentiate (x + 1/x)³ with respect to x.

Answer: dy/dx = 3(x+1/x)²(1 - 1/x²)

Q59. Differentiate (3x + 5)(1 + tan x) with respect to x.

Answer: dy/dx = 3(1+tan x) + (3x+5)sec²x

Q60. Differentiate x² cos x/4 with respect to x.

Answer: dy/dx = (2x cos x - x² sin x)/4

✅ ANSWERS - MCQs

Q.No.	Answer	Q.No.	Answer
1	(a)	11	(b)
2	(a)	12	(d)
3	(a)	13	(a)
4	(b)	14	(a)
5	(b)	15	(a)
6	(c)	16	(b)
7	(c)	17	(b)
8	(d)	18	(b)
9	(b)	19	(b)
10	(d)	20	(b)

📌 KEY FORMULAS SUMMARY

Formula Type	Expression
Limit Definition	lim_x→a f(x) = L
Standard Limit	lim_x→0 sin x/x = 1
Standard Limit	lim_x→0 (1-cos x)/x = 0
Standard Limit	lim_x→0 tan x/x = 1
Standard Limit	lim_x→0 (eˣ-1)/x = 1
Standard Limit	lim_x→0 log(1+x)/x = 1
Derivative Definition	f'(x) = lim_h→0 [f(x+h)-f(x)]/h
Power Rule	d/dx(xⁿ) = nxⁿ⁻¹
Product Rule	d/dx(f·g) = f'g + fg'
Quotient Rule	d/dx(f/g) = (f'g - fg')/g²

🔹 Important Derivatives:

d/dx(sin x) = cos x
d/dx(cos x) = -sin x
d/dx(tan x) = sec² x
d/dx(eˣ) = eˣ
d/dx(logₑ x) = 1/x

📚 Kendriya Vidyalaya Sangathan | Class XI Mathematics
Chapter 13: Limits and Derivatives | Student Support Material

``` --- How to Use This Code on Blogger 1. Login to your Blogger account 2. Go to New Post → Click on HTML View (not Compose) 3. Copy and paste the entire HTML code above 4. Click Publish or Preview to see the formatted content --- What This Code Includes Section Content Introduction Limits, left-hand/right-hand limits, existence of limits Properties of Limits Sum, difference, product, quotient rules Standard Limits Important limits (sin x/x, (1-cos x)/x, tan x/x, eˣ-1/x, etc.) Derivatives Definition, power rule, product rule, quotient rule Derivative Formulas Derivatives of trigonometric, exponential, logarithmic functions MCQ Questions Q1-Q20 with full solutions Case Based Questions Q21-Q25 with all sub-parts Assertion & Reason Q26-Q30 with detailed explanations Very Short Answers Q31-Q45 with complete solutions Short Answers Q46-Q60 with complete solutions Answer Keys All answers in table format Formula Summary All important formulas in one table --- Features · ✅ Complete limit formulas with proper notation · ✅ Standard limits with detailed explanations · ✅ Derivative rules (power, product, quotient) · ✅ Derivatives of trigonometric functions · ✅ Derivatives of exponential and logarithmic functions · ✅ Proper formatting with colored boxes for questions/answers · ✅ Responsive design for mobile/tablet viewing · ✅ Clean textbook-style layout matching the PDF content · ✅ Case studies with practical applications

Q.No.	Answer	Q.No.	Answer
1	(a)	11	(b)
2	(a)	12	(d)
3	(a)	13	(a)
4	(b)	14	(a)
5	(b)	15	(a)
6	(c)	16	(b)
7	(c)	17	(b)
8	(d)	18	(b)
9	(b)	19	(b)
10	(d)	20	(b)

Q.No.	Answer	Q.No.	Answer
1	(a)	11	(b)
2	(a)	12	(d)
3	(a)	13	(a)
4	(b)	14	(a)
5	(b)	15	(a)
6	(c)	16	(b)
7	(c)	17	(b)
8	(d)	18	(b)
9	(b)	19	(b)
10	(d)	20	(b)

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Limits and Derivatives

📐 CHAPTER 13: LIMITS AND DERIVATIVES

📖 INTRODUCTION

🔹 Key Concepts:

🔹 Existence of Limit

📜 PROPERTIES OF LIMITS

📐 STANDARD LIMITS

🔹 Important Limits:

📈 DERIVATIVES

🔹 Derivative Rules

🔹 Sum and Difference Rule

🔹 Product Rule

🔹 Quotient Rule

📝 MCQ QUESTIONS

📚 CASE BASED QUESTIONS

🧠 ASSERTION & REASON QUESTIONS

📝 VERY SHORT ANSWER TYPE QUESTIONS

📝 SHORT ANSWER TYPE QUESTIONS

✅ ANSWERS - MCQs

📌 KEY FORMULAS SUMMARY

🔹 Important Derivatives:

JKAS.IN

Q.No.	Answer	Q.No.	Answer
1	(a)	11	(b)
2	(a)	12	(d)
3	(a)	13	(a)
4	(b)	14	(a)
5	(b)	15	(a)
6	(c)	16	(b)
7	(c)	17	(b)
8	(d)	18	(b)
9	(b)	19	(b)
10	(d)	20	(b)