Class 11 Mathematics Notes Chapter 13 (Chapter 13) – Examplar Problems (English) Book

Alright class, let's get straight into Chapter 13: Limits and Derivatives from your NCERT Exemplar. This chapter is absolutely crucial, not just for your Class 11 exams, but it forms the bedrock of Calculus, which features heavily in many government recruitment exams like NDA, SSC CGL (Tier 2), and others involving quantitative aptitude or mathematics sections. Pay close attention!

Chapter 13: Limits and Derivatives - Detailed Notes (Exemplar Focused)

1. Introduction to Limits

• Concept: A limit describes the value that a function approaches as the input (or index) approaches some value. It's about the neighbourhood of a point, not necessarily the value at the point itself.
• Notation: lim (x→a) f(x) = L (Read as: the limit of f(x) as x approaches 'a' is L).
• Left-Hand Limit (LHL): The value f(x) approaches as x approaches 'a' from the left side (values less than 'a').
  • Notation: lim (x→a⁻) f(x) or lim (h→0) f(a-h)
• Right-Hand Limit (RHL): The value f(x) approaches as x approaches 'a' from the right side (values greater than 'a').
  • Notation: lim (x→a⁺) f(x) or lim (h→0) f(a+h)
• Existence of a Limit: A limit lim (x→a) f(x) exists if and only if LHL = RHL.
  • lim (x→a⁻) f(x) = lim (x→a⁺) f(x) = L (where L is a finite value).
  • Exemplar often tests existence using piecewise functions, modulus functions, or greatest integer functions where LHL and RHL might differ.

2. Algebra of Limits

Assuming lim (x→a) f(x) = L and lim (x→a) g(x) = M (where L and M are finite):

• Sum Rule: lim (x→a) [f(x) + g(x)] = L + M
• Difference Rule: lim (x→a) [f(x) - g(x)] = L - M
• Product Rule: lim (x→a) [f(x) * g(x)] = L * M
• Constant Multiple Rule: lim (x→a) [k * f(x)] = k * L (where k is a constant)
• Quotient Rule: lim (x→a) [f(x) / g(x)] = L / M, provided M ≠ 0.

3. Evaluating Limits

• Direct Substitution: For polynomial functions and rational functions (where the denominator is non-zero at x=a), simply substitute x = a.
  • Example: lim (x→2) (x² + 3) = 2² + 3 = 7
• Indeterminate Forms: If direct substitution results in forms like 0/0, ∞/∞, ∞ - ∞, 0 × ∞, 1^∞, 0⁰, ∞⁰, these are indeterminate forms. We need other methods.
• Factorization Method (for 0/0 form): Factorize the numerator and denominator and cancel the common factor (x-a).
  • Example: lim (x→1) (x² - 1)/(x - 1) = lim (x→1) (x-1)(x+1)/(x-1) = lim (x→1) (x+1) = 1+1 = 2
• Rationalization Method (for 0/0 form involving square roots): Multiply the numerator and denominator by the conjugate of the term containing the square root.
  • Example: lim (x→0) (√(1+x) - 1)/x = lim (x→0) [(√(1+x) - 1)/x] * [(√(1+x) + 1)/(√(1+x) + 1)] = lim (x→0) (1+x - 1) / [x(√(1+x) + 1)] = lim (x→0) x / [x(√(1+x) + 1)] = lim (x→0) 1 / (√(1+x) + 1) = 1 / (√1 + 1) = 1/2
• Using Standard Limits: Memorize these standard results. They are frequently used.

4. Standard Limits (Essential for Competitive Exams)

• lim (x→0) sin(x) / x = 1 (x must be in radians)

• lim (x→0) tan(x) / x = 1 (x must be in radians)

• lim (x→0) (1 - cos(x)) / x = 0

• lim (x→0) (1 - cos(x)) / x² = 1/2

• lim (x→0) sin⁻¹(x) / x = 1

• lim (x→0) tan⁻¹(x) / x = 1

• lim (x→a) (xⁿ - aⁿ) / (x - a) = n * aⁿ⁻¹

• lim (x→0) (eˣ - 1) / x = 1

• lim (x→0) (aˣ - 1) / x = logₑ(a) (or ln a)

• lim (x→0) (1 + x)¹ᐟˣ = e

• lim (x→∞) (1 + 1/x)ˣ = e

• lim (x→0) logₑ(1 + x) / x = 1 (or ln(1+x)/x)

• Note on L'Hôpital's Rule (Useful Shortcut for MCQs): If lim (x→a) f(x)/g(x) is in the 0/0 or ∞/∞ form, then lim (x→a) f(x)/g(x) = lim (x→a) f'(x)/g'(x), provided the latter limit exists. Use with caution, ensure you know the standard methods first.

5. Introduction to Derivatives

• Concept: The derivative represents the instantaneous rate of change of a function. Geometrically, it's the slope of the tangent line to the function's graph at a specific point.
• Derivative at a Point (First Principle): The derivative of f(x) at x = a is:
  • f'(a) = lim (h→0) [f(a+h) - f(a)] / h
• Derivative Function (First Principle): The derivative function f'(x) is:
  • f'(x) = lim (h→0) [f(x+h) - f(x)] / h (provided the limit exists)
• Notation: dy/dx, y', f'(x), D(f(x)) all denote the derivative of y = f(x) with respect to x.
• Differentiability: A function is differentiable at a point 'a' if the limit defining f'(a) exists (i.e., LHD = RHD, where LHD and RHD are Left Hand Derivative and Right Hand Derivative). Differentiability implies continuity, but continuity does not necessarily imply differentiability (e.g., y = |x| at x=0).

6. Algebra of Derivatives

Let u = f(x) and v = g(x) be differentiable functions.

• Sum/Difference Rule: d/dx (u ± v) = du/dx ± dv/dx
• Product Rule (Leibniz Rule): d/dx (u * v) = u * (dv/dx) + v * (du/dx)
• Constant Multiple Rule: d/dx (k * u) = k * (du/dx) (where k is a constant)
• Quotient Rule: d/dx (u / v) = [v * (du/dx) - u * (dv/dx)] / v², provided v ≠ 0.

7. Standard Derivatives (Memorize These!)

• d/dx (c) = 0 (where c is a constant)
• 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 (eˣ) = eˣ
• d/dx (aˣ) = aˣ logₑ(a) (or aˣ ln a)
• d/dx (logₑ x) = 1/x (or d/dx (ln x) = 1/x)
• d/dx (logₐ x) = 1 / (x logₑ a)

Key Focus Areas from Exemplar for Exams:

• Evaluating limits requiring algebraic manipulation beyond simple factorization/rationalization.
• Limits involving trigonometric identities and standard limits.
• Checking the existence of limits for piecewise, modulus ([x]), and greatest integer ({x}) functions.
• Finding derivatives using the first principle (can be lengthy, but conceptually important).
• Applying product and quotient rules diligently, especially with trigonometric functions.
• Understanding the relationship between continuity and differentiability.

Multiple Choice Questions (MCQs)

Here are 10 MCQs covering concepts from this chapter, similar to what you might encounter:

1. The value of lim (x→0) (sin 4x) / (tan 2x) is:
   (A) 1
   (B) 2
   (C) 1/2
   (D) 4

2. If f(x) = (x² - 4) / (x - 2) for x ≠ 2 and f(2) = k, for the function to be continuous at x = 2, the value of k must be:
   (A) 0
   (B) 2
   (C) 4
   (D) Undefined

3. The value of lim (x→π/2) (1 - sin x) / cos²x is:
   (A) 1
   (B) 0
   (C) 1/2
   (D) 2

4. If y = x³ sin x, then dy/dx is:
   (A) 3x² cos x
   (B) x³ cos x + 3x² sin x
   (C) 3x² sin x - x³ cos x
   (D) 3x sin x

5. The derivative of f(x) = |x - 1| at x = 1 is:
   (A) 1
   (B) -1
   (C) 0
   (D) Does not exist

6. The value of lim (x→0) (e^(sin x) - 1) / x is:
   (A) 1
   (B) e
   (C) 0
   (D) Does not exist

7. If f(x) = (ax + b) / (cx + d), then f'(x) is:
   (A) (ad - bc) / (cx + d)²
   (B) (bc - ad) / (cx + d)²
   (C) ad / c
   (D) (acx + bd) / (cx + d)²

8. lim (x→∞) (3x² + 2x - 1) / (5x² - 3x + 2) is equal to:
   (A) 3/5
   (B) 2/(-3)
   (C) -1/2
   (D) ∞

9. The derivative of tan(√x) with respect to x is:
   (A) sec²(√x)
   (B) sec²(√x) / (2√x)
   (C) sec(√x) tan(√x) / (2√x)
   (D) sec²(√x) * (√x)

10. Let f(x) = x if x ≥ 0 and f(x) = -x if x < 0. Then lim (x→0) f(x) is:
    (A) 1
    (B) -1
    (C) 0
    (D) Does not exist

Answers to MCQs:

1. (B) 2
2. (C) 4
3. (C) 1/2
4. (B) x³ cos x + 3x² sin x
5. (D) Does not exist
6. (A) 1
7. (A) (ad - bc) / (cx + d)²
8. (A) 3/5
9. (B) sec²(√x) / (2√x)
10. (C) 0

Study these notes thoroughly. Practice problems from the Exemplar book, focusing on understanding the 'why' behind each step, especially for limit evaluations and derivative rules. Good luck with your preparation!