Limits and Continuity Practice Paper 1

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Mathematical graph showing limit concept

📚 Part 1: Key Concepts and Definitions

Success Criteria: By the end of this section, you will be able to identify limits, understand continuity, and apply basic limit laws.

1. The limit of f(x) as x approaches 3 is 7. This means:

f(3) = 7

f(x) gets closer to 7 as x gets closer to 3

f(x) = 7 for all values of x

x = 3 when f(x) = 7

2. A function is continuous at x = a if (check all that apply):

f(a) exists

lim(x→a) f(x) exists

lim(x→a) f(x) = f(a)

f(x) has no breaks at x = a

3. Key Formula: Limit Laws

If lim(x→a) f(x) = L and lim(x→a) g(x) = M, then:

• Sum Rule: lim(x→a) [f(x) + g(x)] = _______ + _______

• Product Rule: lim(x→a) [f(x) × g(x)] = _______ × _______

• Quotient Rule: lim(x→a) [f(x) ÷ g(x)] = _______ ÷ _______ (if M ≠ 0)

4. Types of discontinuities include:

Removable (hole)

Jump discontinuity

Infinite discontinuity (vertical asymptote)

Oscillating discontinuity

✏️ Part 2: Problem Solving

Success Criteria: You will calculate limits algebraically and graphically, identify points of discontinuity.

5. Calculate lim(x→2) (x² - 4)/(x - 2). Show your work:

Hint for differentiation: Factor the numerator first. Advanced learners: Try L'Hôpital's rule as an alternative method.

6. For f(x) = {x + 1 if x < 2; 5 if x = 2; x² - 1 if x > 2}, find:

a) lim(x→2⁻) f(x) = _______

b) lim(x→2⁺) f(x) = _______

c) f(2) = _______

d) Is f(x) continuous at x = 2? Explain:

7. Sketch a function that has a removable discontinuity at x = 1:

Extension Activity: Advanced learners should also sketch the continuous function that "removes" this discontinuity.

🎯 Part 3: Application and Reflection

Success Criteria: You will apply limit concepts to real-world problems and reflect on your understanding.

8. A ball is thrown upward. Its height h(t) = -5t² + 20t + 1 metres after t seconds. Find the instantaneous velocity at t = 2 seconds using the limit definition:

v(2) = lim(h→0) [h(2+h) - h(2)]/h

Differentiation Support: Start by substituting the values, then expand and simplify step by step.

9. Real-world connection: Explain how continuity relates to a practical situation (e.g., temperature changes, population growth, speed of a car):

10. Self-assessment: Rate your confidence (1-5) and explain one concept you'd like to review:

Confidence level: 1 __ 2 __ 3 __ 4 __ 5

Dyslexia-Friendly Note: This worksheet uses clear fonts, consistent spacing, and bullet points for easier reading. Key terms are in bold for emphasis.

Extension Challenge: Research epsilon-delta definition of limits and write a brief explanation of how it provides a rigorous foundation for calculus.