Unit 1: Limits and Continuity

Master the foundation of calculus with limits and continuity

← Back to Units

📋 Unit Overview

Unit 1 establishes the foundation for all of calculus. You'll learn how to:

  • Evaluate limits using various methods
  • Understand one-sided and two-sided limits
  • Determine continuity of functions
  • Apply the Intermediate Value Theorem
  • Use limits to find asymptotes
💡 Key Insight: Limits are the bridge between algebra and calculus. They allow us to study behavior as we approach specific values, even when direct substitution isn't possible.

🔍 Understanding Limits

What is a Limit?

A limit describes the value that a function approaches as the input approaches a certain value. We write:

limx→a f(x) = L

"The limit of f(x) as x approaches a equals L"

🎯 Key Concept

Limits don't care about what happens AT the point x = a, only what happens NEAR it.

📝 Example 1.1: Basic Limit

Find limx→2 (x² + 3)

Solution: As x approaches 2, x² approaches 4, so the limit is 4 + 3 = 7

Methods for Evaluating Limits

1. Direct Substitution

Try plugging in the value directly first.

limx→3 (2x + 1) = 2(3) + 1 = 7 ✓

2. Factoring

When you get 0/0, try factoring to cancel common terms.

limx→2 (x²-4)/(x-2) = limx→2 (x+2) = 4

3. Rationalizing

For expressions with radicals, multiply by conjugate.

4. Special Limits

Memorize key limits like limx→0 sin(x)/x = 1

⬅️➡️ One-Sided Limits

One-sided limits approach a value from only one direction:

limx→a⁻ f(x)

Left-hand limit (approaching from below)

limx→a⁺ f(x)

Right-hand limit (approaching from above)

🔑 Important Rule

A two-sided limit exists if and only if both one-sided limits exist and are equal.

limx→a f(x) = L ⇔ limx→a⁻ f(x) = limx→a⁺ f(x) = L

📝 Example 1.2: One-Sided Limits

For f(x) = |x|/x, find limx→0 f(x)

Solution:

  • limx→0⁻ f(x) = limx→0⁻ (-x)/x = -1
  • limx→0⁺ f(x) = limx→0⁺ x/x = 1
  • Since -1 ≠ 1, limx→0 f(x) does not exist

🔗 Continuity

Definition of Continuity

A function f(x) is continuous at x = a if:

1. f(a) is defined
2. limx→a f(x) exists
3. limx→a f(x) = f(a)
🎯 Visual Understanding

A function is continuous if you can draw its graph without lifting your pencil. No jumps, breaks, or holes!

📝 Example 1.3: Checking Continuity

Is f(x) = x² continuous at x = 2?

Solution:

  1. f(2) = 4 ✓ (defined)
  2. limx→2 x² = 4 ✓ (exists)
  3. limx→2 x² = f(2) ✓ (equal)

Answer: Yes, f(x) is continuous at x = 2

Types of Discontinuities

Type Description Example
Removable Hole in the graph f(x) = (x²-1)/(x-1) at x = 1
Jump Function "jumps" to different value f(x) = |x|/x at x = 0
Infinite Function approaches ±∞ f(x) = 1/x at x = 0

⭐ Special Limits to Memorize

limx→0 sin(x)/x = 1

Sine limit

limx→0 (1-cos(x))/x = 0

Cosine limit

limx→∞ (1+1/x)ˣ = e

Natural number e

limx→0 (eˣ-1)/x = 1

Exponential limit

🧮 Practice Problems

Problem 1

Find limx→3 (x²-9)/(x-3)

Solution:

This gives us 0/0 when we substitute x = 3, so we need to factor:

(x²-9)/(x-3) = (x+3)(x-3)/(x-3) = x+3

Therefore, limx→3 (x²-9)/(x-3) = limx→3 (x+3) = 6

Problem 2

Is the function f(x) = 1/x continuous at x = 0? Explain.

Solution:

No, f(x) = 1/x is NOT continuous at x = 0 because:

  1. f(0) is undefined (division by zero)
  2. limx→0 1/x does not exist (approaches ±∞)

This is an infinite discontinuity.

Problem 3

Find limx→0 sin(3x)/x

Solution:

We can use the special limit limx→0 sin(x)/x = 1

Let u = 3x, then as x → 0, u → 0

limx→0 sin(3x)/x = limu→0 sin(u)/(u/3) = 3 × limu→0 sin(u)/u = 3 × 1 = 3

💡 Exam Tips

🎯 Multiple Choice
  • Always try direct substitution first
  • Look for patterns in answer choices
  • Use calculator to check your work
  • Remember special limits
✍️ Free Response
  • Show all steps clearly
  • State which method you're using
  • Check your answer makes sense
  • Use proper limit notation
⚠️ Common Mistakes:
  • Forgetting to check one-sided limits
  • Not factoring when you get 0/0
  • Confusing continuity with differentiability
  • Forgetting special limits

🚀 Ready to Test Your Knowledge?

Take our comprehensive Unit 1 practice test with 15 multiple choice questions and 3 free response problems!

📊
15 MC Questions

Progressive difficulty levels

✍️
3 Free Response

Show your work

⏱️
45 Min Timer

Real exam conditions

🎯 Start Practice Test Now

Get instant feedback and detailed solutions!

➡️ What's Next?

Ready for the next challenge?

In Unit 2, you'll learn about differentiation and its fundamental properties.

Go to Unit 2: Differentiation →