Limit Laws in Calculus

Introduction

• Purpose: Learn limit laws to simplify limit calculations algebraically.
• Key Notations:
  • C: A constant (any number).
  • f(x), g(x): Functions where:
    • limit as x -> a of f(x) = F
    • limit as x -> a of g(x) = G

Limit Laws

1. Sum Law

• Statement: limit as x -> a of [f(x) + g(x)] = limit as x -> a of f(x) + limit as x -> a of g(x)
• Example:
  • Given: limit as x -> 2 of [x^2 + x + x + 1]
  • Split: limit as x -> 2 of (x^2 + x) + limit as x -> 2 of (x + 1)
  • Calculate individually: 4 + 2 + 3 = 9
  • Combined function: limit as x -> 2 of [x^2 + 2x + 1] = 4 + 4 + 1 = 9
  • Result: Both methods give 9

2. Multiplication by a Constant Law

• Statement: limit as x -> a of [C * f(x)] = C * limit as x -> a of f(x)
• Example:
  • Given: limit as x -> 2 of [3x^2 + 3]
  • Direct Evaluation: 3 * 4 + 3 = 15
  • Pull out constant: 3 * (limit as x -> 2 of x^2 + 1)
  • Simplify: 3 * 5 = 15
  • Result: Both methods give 15*

3. Product Law

• Statement: limit as x -> a of [f(x) * g(x)] = limit as x -> a of f(x) * limit as x -> a of g(x)
• Example:
  • Given: limit as x -> 2 of [x^2 + 2x + 1]
  • Factor: limit as x -> 2 of [(x + 1) * (x + 1)]
  • Split: limit as x -> 2 of (x + 1) * limit as x -> 2 of (x + 1)
  • Evaluate: 3 * 3 = 9
  • Result: Both methods give 9*

4. Division Law

• Statement: limit as x -> a of [f(x) / g(x)] = limit as x -> a of f(x) / limit as x -> a of g(x) provided limit as x -> a of g(x) != 0
• Example:
  • Given: limit as x -> 2 of [3x^2 / (x^2 + 1)]
  • Direct Evaluation: 6 / 5
  • Split: limit as x -> 2 of 3x / limit as x -> 2 of (x^2 + 1)
  • Evaluate: 6 / 5
  • Result: Both methods give 6 / 5

Practice Question

• Given: limit as x -> 5 of [10x^2 + 5x - 20]
• Use multiplication law:
  • Pull out factor: 5 * limit as x -> 5 of [2x^2 + x - 4]
• Use sum law:
  • Split: 5 * (limit as x -> 5 of 2x^2 + limit as x -> 5 of x + limit as x -> 5 of -4)
  • Individual Limits:
    • 2 * limit as x -> 5 of x^2 = 50
    • limit as x -> 5 of x = 5
    • limit as x -> 5 of -4 = -4
  • Sum: 50 + 5 - 4 = 51
  • Multiply: 5 * 51 = 255
• Direct Evaluation Comparison:
  • 10 * 25 + 5 * 5 - 20 = 255

Conclusion

• Limit laws simplify complex limit calculations.
• More proofs will be covered in advanced courses.
• Practice and apply these laws to save time and ensure accuracy in exams.