📐 Concept diagram

04-01 - Limits

Phase: 4 | Subject: 04-01 Prerequisites: 03-05-exponential-and-logarithmic-functions.md, 03-07-sequences-and-series.md Next subject: 04-02-continuity.md

Learning Objectives

By the end of this subject, you will be able to:

1. Understand the intuitive concept of a limit
2. Evaluate limits using direct substitution
3. Handle indeterminate forms algebraically
4. Understand one-sided limits
5. Apply the squeeze theorem

Core Content

What is a Limit?

⚠️ THIS IS CRITICAL — limits are the foundation of ALL calculus. Every concept from derivatives to integrals to infinite series builds on limits. Master this now.

The limit of f(x) as x approaches a is the value that f(x) gets arbitrarily close to (but may not actually reach).

Notation: $lim(x→a)$ f(x) = L

Intuition: As x gets closer and closer to a from either side, f(x) gets closer and closer to L.

Example: $lim(x→2)$ (x² + 1) = 5 As x approaches 2, x² + 1 approaches 2² + 1 = 5.

Key: The limit doesn't care what happens AT x = a, only what happens NEAR x = a.

The Epsilon-Delta Definition (Conceptual)

For those interested in the formal definition: we say $lim(x→a)$ f(x) = L if, for any tiny "error tolerance" ε > 0, we can find a δ > 0 such that whenever 0 < |x - a| < δ (x is within δ of a, but not equal to a), we have |f(x) - L| < ε (f(x) is within ε of L). In words: you can get f(x) as close as you want to L by making x close enough to a. This rigorous definition underpins all of calculus, but for practical limit evaluation, the algebraic methods above suffice.

Evaluating Limits by Direct Substitution

If f(x) is continuous at x = a: $lim(x→a)$ f(x) = f(a)

Example: $lim(x→3)$ (2x + 5) = 2(3) + 5 = 11

Indeterminate Forms

When direct substitution gives 0/0 or ∞/∞, we need algebraic manipulation.

Factoring

Example: $lim(x→2)$ (x² - 4)/(x - 2) Direct: (4 - 4)/(2 - 2) = 0/0 — indeterminate!

Factor numerator: (x - 2)(x + 2)/(x - 2) = x + 2 (for x ≠ 2) $lim(x→2)$ (x + 2) = 4

Rationalising

Example: $lim(x→4)$ (√x - 2)/(x - 4) Direct: (2 - 2)/(4 - 4) = 0/0

Multiply by conjugate: (√x - 2)/(x - 4) × (√x + 2)/(√x + 2) = (x - 4)/((x - 4)(√x + 2)) = 1/(√x + 2)

$lim(x→4)$ 1/(√x + 2) = 1/(2 + 2) = 1/4

Combining Fractions

Example: $lim(x→2)$ (1/(x-1) - 1/(x+1)) = $lim(x→2)$ [(x+1) - (x-1)] / [(x-1)(x+1)] = $lim(x→2)$ (x+1-x+1)/((x-1)(x+1)) = $lim(x→2)$ 2/(x² - 1) = 2/(4 - 1) = 2/3

One-Sided Limits

• Left-hand limit: $lim(x→a⁻)$ f(x) — x approaches a from the LEFT (x < a)
• Right-hand limit: $lim(x→a⁺)$ f(x) — x approaches a from the RIGHT (x > a)

For the overall limit to exist, BOTH one-sided limits must exist and be equal.

Example: For f(x) = |x|/x at x = 0: $lim(x→0⁻)$ |x|/x = $lim(x→0⁻)$ (-x)/x = -1 $lim(x→0⁺)$ |x|/x = $lim(x→0⁺)$ x/x = 1

Since -1 ≠ 1, the limit DOES NOT EXIST.

Limits at Infinity

Example: $lim(x→∞)$ (2x² + 3)/(x² - 1) = $lim(x→∞)$ (2 + 3/x²)/(1 - 1/x²) = (2 + 0)/(1 - 0) = 2

Rule: Divide numerator and denominator by the highest power of x.

Squeeze Theorem (Sandwich Theorem)

If g(x) ≤ f(x) ≤ h(x) for all x near a, and: $lim(x→a)$ g(x) = $lim(x→a)$ h(x) = L Then: $lim(x→a)$ f(x) = L

Example: Prove $lim(x→0)$ x²·sin(1/x) = 0

Since -1 ≤ sin(1/x) ≤ 1, multiplying by x² ≥ 0: -x² ≤ x²·sin(1/x) ≤ x²

$lim(x→0)$ (-x²) = $lim(x→0)$ x² = 0 Therefore, $lim(x→0)$ x²·sin(1/x) = 0

Key Terms

• 04 01 Limits
• Combining Fractions
• Correct: B)
• Correct: C)
• Evaluating Limits by Direct Substitution
• Example 1: Factoring
• Example 2: Rationalising
• Example 3: Infinite limit
• Factoring
• Indeterminate Forms
• Limits at Infinity
• One-Sided Limits

Worked Examples

Example 1: Factoring

$lim(x→3)$ (x² - 9)/(x - 3) = $lim(x→3)$ (x - 3)(x + 3)/(x - 3) = $lim(x→3)$ (x + 3) = 6

Example 2: Rationalising

$lim(x→0)$ (√(x + 4) - 2)/x Multiply by conjugate: (√(x+4) - 2)(√(x+4) + 2) / (x(√(x+4) + 2)) = (x + 4 - 4) / (x(√(x+4) + 2)) = x / (x(√(x+4) + 2)) = 1/(√(x+4) + 2)

$lim(x→0)$ = 1/(2 + 2) = 1/4

Example 3: Infinite limit

$lim(x→0⁺)$ 1/x = +∞ $lim(x→0⁻)$ 1/x = -∞ Limit does not exist (different from ±∞).

Quiz

Q1: What does the concept of Combining Fractions primarily refer to in this subject?

A) A computational error related to Combining Fractions B) A visual representation of Combining Fractions C) A historical anecdote about Combining Fractions D) The definition and application of Combining Fractions

Correct: D)

• If you chose A: This is incorrect. Combining Fractions is defined as: the definition and application of combining fractions. The other options describe different aspects that are not the primary focus.
• If you chose B: This is incorrect. Combining Fractions is defined as: the definition and application of combining fractions. The other options describe different aspects that are not the primary focus.
• If you chose C: This is incorrect. Combining Fractions is defined as: the definition and application of combining fractions. The other options describe different aspects that are not the primary focus.
• If you chose D: Combining Fractions is defined as: the definition and application of combining fractions. The other options describe different aspects that are not the primary focus. Correct!

Q2: What is the primary purpose of Evaluating Limits by Direct Substitution?

A) It is used only in advanced research contexts B) It is used to evaluating limits by direct substitution in mathematical analysis C) It is primarily a historical notation system D) It replaces all other methods in this domain

Correct: B)

• If you chose A: This is incorrect. Evaluating Limits by Direct Substitution serves the purpose described in the correct answer. The other options misrepresent its role.
• If you chose B: Evaluating Limits by Direct Substitution serves the purpose described in the correct answer. The other options misrepresent its role. Correct!
• If you chose C: This is incorrect. Evaluating Limits by Direct Substitution serves the purpose described in the correct answer. The other options misrepresent its role.
• If you chose D: This is incorrect. Evaluating Limits by Direct Substitution serves the purpose described in the correct answer. The other options misrepresent its role.

Q3: Which statement about Factoring is TRUE?

A) Factoring is an advanced topic beyond this subject's scope B) Factoring is not related to this subject C) Factoring is a fundamental concept covered in this subject D) Factoring is mentioned only as a historical footnote

Correct: C)

• If you chose A: This is incorrect. Factoring is a fundamental concept covered in this subject. This subject covers Factoring as part of its core content.
• If you chose B: This is incorrect. Factoring is a fundamental concept covered in this subject. This subject covers Factoring as part of its core content.
• If you chose C: Factoring is a fundamental concept covered in this subject. This subject covers Factoring as part of its core content. Correct!
• If you chose D: This is incorrect. Factoring is a fundamental concept covered in this subject. This subject covers Factoring as part of its core content.

Q4: Based on the worked examples in this subject, what is the correct result?

A) The inverse of the correct answer B) A different result from a common mistake C) An unrelated numerical value D) B)**

Correct: D)

• If you chose A: This is incorrect. The worked examples show that the result is B)**. The other options represent common errors.
• If you chose B: This is incorrect. The worked examples show that the result is B)**. The other options represent common errors.
• If you chose C: This is incorrect. The worked examples show that the result is B)**. The other options represent common errors.
• If you chose D: The worked examples show that the result is B)**. The other options represent common errors. Correct!

Q5: How are Factoring and Indeterminate Forms related?

A) Factoring is the inverse of Indeterminate Forms B) Factoring and Indeterminate Forms are completely unrelated topics C) Factoring is a special case of Indeterminate Forms D) Factoring and Indeterminate Forms are closely related concepts

Correct: D)

• If you chose A: This is incorrect. Both Factoring and Indeterminate Forms are covered in this subject as interconnected topics.
• If you chose B: This is incorrect. Both Factoring and Indeterminate Forms are covered in this subject as interconnected topics.
• If you chose C: This is incorrect. Both Factoring and Indeterminate Forms are covered in this subject as interconnected topics.
• If you chose D: Both Factoring and Indeterminate Forms are covered in this subject as interconnected topics. Correct!

Q6: What is a common pitfall when working with Limits at Infinity?

A) A common mistake is confusing Limits at Infinity with a similar concept B) Limits at Infinity is always computed the same way in all contexts C) Limits at Infinity has no common misconceptions D) The main error with Limits at Infinity is using it when it is not needed

Correct: A)

• If you chose A: Students often confuse Limits at Infinity with similar-sounding or related concepts. Pay attention to the precise definitions. Correct!
• If you chose B: This is incorrect. Students often confuse Limits at Infinity with similar-sounding or related concepts. Pay attention to the precise definitions.
• If you chose C: This is incorrect. Students often confuse Limits at Infinity with similar-sounding or related concepts. Pay attention to the precise definitions.
• If you chose D: This is incorrect. Students often confuse Limits at Infinity with similar-sounding or related concepts. Pay attention to the precise definitions.

Q7: When should you apply One-Sided Limits?

A) Apply One-Sided Limits to solve problems in this subject's domain B) Avoid One-Sided Limits unless explicitly instructed C) One-Sided Limits is not practically useful D) Use One-Sided Limits only in pure mathematics contexts

Correct: A)

• If you chose A: One-Sided Limits is a practical tool used throughout this subject to solve relevant problems. Correct!
• If you chose B: This is incorrect. One-Sided Limits is a practical tool used throughout this subject to solve relevant problems.
• If you chose C: This is incorrect. One-Sided Limits is a practical tool used throughout this subject to solve relevant problems.
• If you chose D: This is incorrect. One-Sided Limits is a practical tool used throughout this subject to solve relevant problems.

Practice Problems

1. $lim(x→2)$ (x² - 4)/(x - 2) Answer: Factor: (x-2)(x+2)/(x-2) = x+2. Limit = 4.

2. $lim(x→0)$ (√(x+1) - 1)/x Answer: Multiply by conjugate: (x+1-1)/(x(√(x+1)+1)) = 1/(√(x+1)+1). Limit = 1/2.

3. $lim(x→∞)$ (3x² + 2)/(x² - 1) Answer: Divide by x²: (3 + 2/x²)/(1 - 1/x²) → 3/1 = 3.

4. $lim(x→0⁻)$ 1/x Answer: -∞ (approaches negative infinity from the left).

5. $lim(x→0)$ sin(x)/x Answer: 1 (fundamental trigonometric limit, proven via squeeze theorem).

Summary

Key takeaways:

• Limit: value f(x) approaches as x → a
• Direct substitution works when function is continuous
• 0/0: factor, rationalise, or combine fractions
• One-sided limits: x → a⁻ or x → a⁺
• Overall limit exists iff both one-sided limits equal
• Squeeze theorem: bound f between g and h with same limit

Pitfalls

• Confusing the value at a point with the limit: $lim(x→a)$ f(x) is about what f(x) approaches NEAR a, not the value AT a. A function can have a limit at a point even if f(a) is undefined or different. The classic example: f(x) = (x² - 4)/(x - 2) at x = 2 — f(2) is undefined, but the limit is 4.
• Not checking both one-sided limits: The overall limit exists only if the left-hand limit and right-hand limit are equal. For piecewise functions or functions like |x|/x, always check both sides. If $lim(x→a⁻)$ ≠ $lim(x→a⁺)$, the limit does NOT exist, even if each one-sided limit exists individually.
• Incorrectly cancelling factors in rational expressions: When you factor and cancel (x - a)/(x - a), you're only allowed to do so because x ≠ a in the limit context. After cancellation, the simplified expression is valid for evaluating the limit. But don't forget that the original function is still undefined at x = a — the limit describes the behaviour near a, not at a.
• Dividing by the wrong power at infinity: When evaluating $lim(x→∞)$ of a rational expression, divide numerator and denominator by the HIGHEST power of x that appears. For (3x² + 2)/(x² - 1), divide by x². For (2x + 1)/(x³ + x), divide by x³. Using the wrong highest power leads to incorrect limits (e.g., incorrectly concluding divergence when the limit is 0).
• Assuming a limit always exists: Some limits simply do not exist (e.g., $lim(x→0)$ sin(1/x) oscillates, $lim(x→0)$ 1/x has different ±∞ from each side). The squeeze theorem can help prove existence in some cases, but not all limits can be evaluated — sometimes the correct answer is "does not exist."

Next Steps

Next up: 04-02-continuity.md