Class 11 Mathematics Notes Chapter 13 (Limits and derivatives) – Mathematics Book

Detailed Notes with MCQs of Chapter 13: Limits and Derivatives. This is a foundational chapter for Calculus, and understanding it thoroughly is crucial not just for your Class 11 exams but also forms the bedrock for many quantitative sections in government exams.

We'll break this down into two main parts: Limits and Derivatives.

Part 1: Limits

1. Intuitive Idea of Limits:
Imagine approaching a specific point on a graph from both the left and the right side along the curve. The value (y-value) that the function seems to approach as you get infinitely close to that specific x-value (without necessarily reaching it) is the limit of the function at that point.

• Left-Hand Limit (LHL): The value the function f(x) approaches as x approaches a point 'a' from values less than 'a'. Denoted as: lim (x→a⁻) f(x)
• Right-Hand Limit (RHL): The value the function f(x) approaches as x approaches a point 'a' from values greater than 'a'. Denoted as: lim (x→a⁺) f(x)
• Existence of a Limit: For the limit of f(x) as x approaches 'a' to exist, the LHL must be equal to the RHL, and this value must be finite.
  lim (x→a) f(x) exists if and only if lim (x→a⁻) f(x) = lim (x→a⁺) f(x) = L (where L is a finite number).

2. Algebra of Limits:
If lim (x→a) f(x) = L and lim (x→a) g(x) = M (where L and M are finite), 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
• Scalar 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 = L^n

3. Evaluating Limits:

• Direct Substitution: For polynomials and rational functions, if the function is defined at x = a (i.e., the denominator is not zero for rational functions), simply substitute x = a into the function.
  • Example: lim (x→2) (x² + 3) = 2² + 3 = 7
• Indeterminate Forms (0/0, ∞/∞, etc.): If direct substitution results in an indeterminate form, we need other methods:
  • Factorization: Factorize the numerator and denominator and cancel common factors.
    • 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: If the expression involves square roots, multiply the numerator and denominator by the conjugate.
    • Example: lim (x→0) [√(x+1) - 1] / x (Multiply by [√(x+1) + 1] / [√(x+1) + 1])
  • Using Standard Limits: Memorize and apply standard limit formulas.

4. Standard Limits (Crucial for Exams):

• 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) [ (1 - cos(x)) / x² ] = 1/2
• lim (x→0) [ sin⁻¹(x) / x ] = 1
• lim (x→0) [ tan⁻¹(x) / x ] = 1
• 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 lim (x→0) [ ln(1 + x) / x ] = 1
• lim (x→∞) [ (1 + 1/x)ˣ ] = e
• lim (x→0) [ (1 + x)¹/ˣ ] = e

5. Limits of Polynomial and Rational Functions:

• For a polynomial P(x), lim (x→a) P(x) = P(a).
• For a rational function f(x) = P(x) / Q(x), lim (x→a) f(x) = P(a) / Q(a) provided Q(a) ≠ 0. If Q(a) = 0, check if P(a) is also 0. If P(a) ≠ 0 and Q(a) = 0, the limit does not exist (tends to ±∞). If P(a) = 0 and Q(a) = 0, use factorization or other methods.

Part 2: Derivatives

1. Intuitive Idea of Derivatives:
The derivative of a function f(x) at a point x = a represents:

• The instantaneous rate of change of the function at that point.
• The slope of the tangent line to the curve y = f(x) at the point (a, f(a)).

2. Definition (Derivative from First Principle):
The derivative of a function f(x) with respect to x, denoted as f'(x), dy/dx, or y', is defined as:
f'(x) = lim (h→0) [ (f(x + h) - f(x)) / h ]
provided the limit exists. This is also called the delta method or ab-initio method.

3. Algebra of Derivatives:
If u = f(x) and v = g(x) are differentiable functions:

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

4. Standard Derivatives (Essential Formulas):

• d/dx (c) = 0 (where c is a constant)
• d/dx (xⁿ) = n * xⁿ⁻¹ (Power Rule)
• 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) or aˣ * ln(a)
• d/dx (logₑ x) or d/dx (ln x) = 1/x (for x > 0)
• d/dx (logₐ x) = 1 / (x * logₑ a) (for x > 0, a > 0, a ≠ 1)

5. Derivatives of Polynomial and Trigonometric Functions:
These are found by applying the standard derivative formulas and the algebra of derivatives (sum, difference, product, quotient rules).

• Example: Find the derivative of f(x) = x² * sin x.
  Using the product rule (u = x², v = sin x):
  f'(x) = (d/dx(x²)) * sin x + x² * (d/dx(sin x))
f'(x) = (2x) * sin x + x² * (cos x)
f'(x) = 2x sin x + x² cos x

Key Takeaways for Government Exams:

• Master Standard Limits and Derivatives: Quick recall is essential.
• Understand Indeterminate Forms: Know how to handle 0/0 using factorization, rationalization, or standard limits.
• Apply Algebra Rules: Be proficient with rules for limits and derivatives (sum, product, quotient).
• First Principle: Understand the concept, but direct application is less common in MCQs compared to using rules and formulas. However, definition-based questions can appear.
• Practice: Work through numerous problems to build speed and accuracy.

Multiple Choice Questions (MCQs):

1. The value of lim (x→2) [ (x³ - 8) / (x - 2) ] is:
   (A) 4
   (B) 8
   (C) 12
   (D) 16

2. lim (x→0) [ sin(5x) / x ] is equal to:
   (A) 1
   (B) 5
   (C) 1/5
   (D) 0

3. If f(x) = x³ + 2x² - 5x + 1, then f'(1) is:
   (A) 1
   (B) 2
   (C) 0
   (D) 3

4. The derivative of f(x) = x * cos(x) is:
   (A) cos(x) - x * sin(x)
   (B) sin(x) + x * cos(x)
   (C) cos(x) + x * sin(x)
   (D) -sin(x) - x * cos(x)

5. lim (x→0) [ (e²ˣ - 1) / x ] is equal to:
   (A) 1
   (B) 2
   (C) e
   (D) 1/2

6. The derivative of f(x) = tan(x) / x is:
   (A) (x sec²(x) - tan(x)) / x²
   (B) (x sec²(x) + tan(x)) / x²
   (C) (tan(x) - x sec²(x)) / x²
   (D) sec²(x) / 1

7. Consider the function f(x) = |x| / x for x ≠ 0. What is lim (x→0) f(x)?
   (A) 1
   (B) -1
   (C) 0
   (D) Does not exist

8. The derivative of y = √x + 1/√x is:
   (A) (1/2√x) + (1/2x√x)
   (B) (1/2√x) - (1/2x√x)
   (C) (1/√x) - (1/x√x)
   (D) (1/√x) + (1/x√x)

9. lim (x→π/2) [ (1 - sin x) / cos²x ] is equal to:
   (A) 1
   (B) 0
   (C) 1/2
   (D) 2

10. The derivative of a constant function is always:
    (A) 1
    (B) The constant itself
    (C) 0
    (D) Undefined

Answer Key:

1. (C) [Use (xⁿ - aⁿ)/(x-a) formula or factorize x³-8]
2. (B) [Multiply and divide by 5: lim (x→0) [ 5 * sin(5x) / (5x) ] = 5 * 1 = 5]
3. (B) [f'(x) = 3x² + 4x - 5. f'(1) = 3(1)² + 4(1) - 5 = 3 + 4 - 5 = 2]
4. (A) [Product rule: d/dx(x)*cos(x) + x*d/dx(cos(x)) = 1*cos(x) + x*(-sin(x)) = cos(x) - x sin(x)]
5. (B) [Multiply and divide by 2: lim (x→0) [ 2 * (e²ˣ - 1) / (2x) ] = 2 * 1 = 2]
6. (A) [Quotient rule: ( (d/dx(tanx))*x - tanx*(d/dx(x)) ) / x² = (sec²(x)*x - tan(x)*1) / x²]
7. (D) [LHL = lim (x→0⁻) (-x/x) = -1. RHL = lim (x→0⁺) (x/x) = 1. Since LHL ≠ RHL, the limit does not exist.]
8. (B) [y = x^(1/2) + x^(-1/2). dy/dx = (1/2)x^(-1/2) + (-1/2)x^(-3/2) = (1/2√x) - (1/2x√x)]
9. (C) [cos²x = 1 - sin²x = (1-sinx)(1+sinx). Limit becomes lim (x→π/2) [ (1 - sin x) / ((1-sinx)(1+sinx)) ] = lim (x→π/2) [ 1 / (1+sinx) ] = 1 / (1+sin(π/2)) = 1 / (1+1) = 1/2]
10. (C) [The rate of change of a constant is zero.]

Study these notes carefully, paying special attention to the standard limits and derivative formulas. Practice evaluating limits and finding derivatives using the rules. Good luck with your preparation!