Class 11 Mathematics Notes Chapter 18 (Chapter 18) – Examplar Problems (English) Book

Alright students, let's get straight into Chapter 18: Limits and Derivatives. This is a cornerstone chapter, forming the very bedrock of Calculus. Understanding these concepts thoroughly is crucial, not just for your Class 11 exams, but significantly for competitive government exams where Calculus questions are frequently asked. The NCERT Exemplar problems often test a deeper understanding, so pay close attention.

Chapter 18: Limits and Derivatives - Detailed Notes

Part 1: Limits

1. Intuitive Idea of Limit:

   • The limit of a function f(x) as x approaches a value a (denoted as lim x→a f(x) = L) represents the value that f(x) gets arbitrarily close to as x gets closer and closer to a, without necessarily being equal to a.
   • The function need not be defined at x = a for the limit to exist.
   • Example: Consider f(x) = (x² - 4) / (x - 2). We can't substitute x = 2 directly (0/0 form). But for x ≠ 2, f(x) = (x-2)(x+2) / (x-2) = x + 2. As x gets very close to 2 (like 1.9, 1.99, or 2.1, 2.01), f(x) gets very close to 4. So, lim x→2 f(x) = 4.

2. Indeterminate Forms:

   • Limits are particularly useful when direct substitution leads to undefined or indeterminate forms like: 0/0, ∞/∞, ∞ - ∞, 0 × ∞, 1^∞, 0^0, ∞^0.
   • The most common forms you'll encounter initially are 0/0 and ∞/∞.

3. Left-Hand Limit (LHL) and Right-Hand Limit (RHL):

   • LHL: The value f(x) approaches as x approaches a from the left side (values less than a). Denoted as lim x→a⁻ f(x) or lim h→0 f(a - h).
   • RHL: The value f(x) approaches as x approaches a from the right side (values greater than a). Denoted as lim x→a⁺ f(x) or lim h→0 f(a + h).
   • Existence of Limit: The limit lim x→a f(x) exists if and only if LHL = RHL. If they are not equal, the limit does not exist at x = a. This is crucial for piecewise functions and functions involving modulus or greatest integer functions.

4. Algebra of Limits:
   If lim x→a f(x) = L and lim x→a g(x) = M, then:

   • 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)
   • Power Rule: lim x→a [f(x)]ⁿ = Lⁿ

5. Methods for Evaluating Limits (Especially for 0/0 form):

   • Direct Substitution: If f(a) is defined and not indeterminate, then lim x→a f(x) = f(a). Always try this first.
   • Factorization: Useful for rational functions (polynomials/polynomials). Factorize the numerator and denominator and cancel common factors.
   • Rationalization: Used when expressions involve square roots. Multiply and divide by the conjugate expression.
   • Using Standard Limits: Memorize these thoroughly! They are essential for quick evaluation.
     • lim x→a (xⁿ - aⁿ) / (x - a) = n aⁿ⁻¹
     • 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 (eˣ - 1) / x = 1
     • lim x→0 (aˣ - 1) / x = logₑ(a) (or ln a)
     • lim x→0 logₑ(1 + x) / x = 1 (or ln(1+x)/x)
     • Important Variations: lim x→0 sin(kx) / x = k, lim x→0 tan(kx) / x = k, lim x→0 (1 - cos x) / x² = 1/2.
   • Limits at Infinity (x → ∞ or x → -∞): For rational functions, divide the numerator and the denominator by the highest power of x present in the denominator. Then use the fact that lim x→∞ (c/xⁿ) = 0 for n > 0.

6. Sandwich Theorem (Squeeze Principle):

   • If g(x) ≤ f(x) ≤ h(x) for all x in the neighbourhood of a (except possibly at a), and if lim x→a g(x) = L and lim x→a h(x) = L, then lim x→a f(x) = L.
   • Useful for limits involving trigonometric functions like x sin(1/x).

Part 2: Derivatives

1. Concept of Derivative:

   • Geometrically: The derivative of f(x) at a point x = a, denoted f'(a), represents the slope of the tangent line to the curve y = f(x) at the point (a, f(a)).
   • Physically: It represents the instantaneous rate of change of the function with respect to its variable. (e.g., velocity is the derivative of displacement).

2. Derivative using First Principle (Definition or ab-initio method):

   • The derivative of a function f(x) is defined as:
     f'(x) = lim h→0 [f(x + h) - f(x)] / h
   • Alternatively, the derivative at a specific point x = a is:
     f'(a) = lim x→a [f(x) - f(a)] / (x - a)
   • You must know how to use this definition to find derivatives of simple functions (like xⁿ, sin x, cos x, eˣ, constants).

3. Differentiability:

   • A function f(x) is said to be differentiable at x = a if the limit lim h→0 [f(a + h) - f(a)] / h exists (i.e., the Left-Hand Derivative and Right-Hand Derivative exist and are equal).
   • Key Relationship: If a function is differentiable at a point, it must be continuous at that point.
   • Important Converse: Continuity does not imply differentiability. A classic example is f(x) = |x| which is continuous at x = 0 but not differentiable at x = 0 (sharp corner).

4. Algebra of Derivatives:
   If u(x) and v(x) are differentiable functions:

   • Sum/Difference Rule: d/dx [u(x) ± v(x)] = u'(x) ± v'(x)
   • Product Rule: d/dx [u(x) * v(x)] = u'(x)v(x) + u(x)v'(x)
   • Quotient Rule: d/dx [u(x) / v(x)] = [u'(x)v(x) - u(x)v'(x)] / [v(x)]² (where v(x) ≠ 0)
   • Constant Multiple Rule: d/dx [k * u(x)] = k * u'(x)

5. Standard Derivatives (Formulas to Memorize):

   • d/dx (xⁿ) = n xⁿ⁻¹ (Power Rule)
   • d/dx (constant) = 0
   • d/dx (sin x) = cos x
   • d/dx (cos x) = -sin x
   • d/dx (tan x) = sec² x
   • d/dx (cot x) = -csc² x
   • d/dx (sec x) = sec x tan x
   • d/dx (csc x) = -csc x cot x
   • d/dx (eˣ) = eˣ
   • d/dx (aˣ) = aˣ logₑ(a)
   • d/dx (logₑ x) = 1/x (for x > 0)
   • d/dx (logₐ x) = 1 / (x logₑ a) (for x > 0)

6. Derivatives of Polynomial and Trigonometric Functions:

   • Use the standard formulas combined with the algebra of derivatives (sum, difference, product, quotient rules) to find derivatives of more complex functions built from these basic blocks.

Tips for Government Exams & Exemplar Focus:

• Master Standard Limits & Derivatives: Non-negotiable. Speed and accuracy depend on this.
• Practice Indeterminate Forms: Be comfortable with factorization, rationalization, and applying standard limits creatively.
• Understand LHL/RHL: Crucial for piecewise functions and checking the existence of limits.
• First Principle: Understand the concept and be able to apply it for basic functions as sometimes questions are framed around the definition itself.
• Differentiability vs. Continuity: Understand the implication and the counterexample (|x|).
• Algebra of Derivatives: Apply product and quotient rules carefully, avoiding sign errors.
• Exemplar Problems: Work through them diligently. They often involve slightly more complex algebra or require combining multiple concepts.

Now, let's test your understanding with some multiple-choice questions.

Multiple Choice Questions (MCQs)

1. The value of lim x→0 (sin 4x) / (sin 2x) is:
   (a) 1
   (b) 2
   (c) 1/2
   (d) 0

2. lim x→π (sin x) / (x - π) is equal to:
   (a) 1
   (b) -1
   (c) π
   (d) 0

3. If f(x) = (x² - 9) / (x - 3), then lim x→3 f(x) is:
   (a) 0
   (b) 3
   (c) 6
   (d) Does not exist

4. The value of lim x→0 (e^(2x) - 1) / x is:
   (a) 1
   (b) 2
   (c) 1/2
   (d) e

5. If f(x) = x sin x, then f'(π/2) is:
   (a) 0
   (b) 1
   (c) π/2
   (d) -1

6. The derivative of f(x) = x³ - 2x² + 5 at x = -1 is:
   (a) -7
   (b) 7
   (c) 3
   (d) 1

7. lim x→∞ (3x² + 2x - 1) / (5x² - 3x + 2) is equal to:
   (a) 3/5
   (b) 2/-3
   (c) -1/2
   (d) ∞

8. The derivative of y = tan x / x is:
   (a) (x sec² x - tan x) / x²
   (b) (x sec² x + tan x) / x²
   (c) sec² x / 1
   (d) (tan x - x sec² x) / x²

9. Consider the function f(x) = |x - 1|. At x = 1, the function is:
   (a) Differentiable and Continuous
   (b) Continuous but not Differentiable
   (c) Differentiable but not Continuous
   (d) Neither Continuous nor Differentiable

10. Using the first principle, the derivative of f(x) = cos x is:
    (a) sin x
    (b) -sin x
    (c) cos x
    (d) -cos x

Answers to MCQs:

1. (b) 2
2. (b) -1 [Hint: Let y = x - π, as x→π, y→0. lim y→0 sin(π+y)/y = lim y→0 (-sin y)/y = -1]
3. (c) 6 [Hint: Factorize x²-9]
4. (b) 2 [Hint: Use standard limit lim x→0 (e^(kx)-1)/x = k]
5. (b) 1 [Hint: Use product rule: f'(x) = 1sin x + xcos x. f'(π/2) = sin(π/2) + (π/2)cos(π/2) = 1 + 0 = 1]
6. (b) 7 [Hint: f'(x) = 3x² - 4x. f'(-1) = 3(-1)² - 4(-1) = 3 + 4 = 7]
7. (a) 3/5 [Hint: Divide numerator and denominator by x²]
8. (a) (x sec² x - tan x) / x² [Hint: Use quotient rule]
9. (b) Continuous but not Differentiable [Hint: Graph has a sharp corner at x=1]
10. (b) -sin x [Hint: Apply the definition lim h→0 (cos(x+h) - cos x)/h]

Study these notes carefully and practice a wide variety of problems from your NCERT book and the Exemplar. Good luck!