📐 Concept diagram

04-09 - L'Hôpital's Rule

Phase: 4 | Subject: 04-09 Prerequisites: 04-01-limits.md (indeterminate forms), 04-05-derivatives-of-elementary-functions.md Next subject: 04-10-newtons-method.md

Learning Objectives

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

1. Recognise indeterminate forms 0/0 and ∞/∞
2. Apply L'Hôpital's Rule
3. Handle repeated applications of L'Hôpital's Rule
4. Deal with other indeterminate forms (0·∞, ∞-∞, 0⁰, 1^∞, ∞⁰)

Core Content

Indeterminate Forms

When evaluating limits, direct substitution may give: - 0/0 - ∞/∞ - 0·∞ - ∞ - ∞ - 0⁰, 1^∞, ∞⁰

These are NOT actual values — they're "competitions" between functions.

L'Hôpital's Rule

⚠️ THIS IS CRITICAL — L'Hôpital's Rule is a powerful shortcut, but ONLY for 0/0 or ∞/∞ forms. Do NOT apply it to limits that are not indeterminate, and always check that f' and g' exist near a with g'(x) ≠ 0.

If $lim(x→a)$ f(x) = 0 and $lim(x→a)$ g(x) = 0 (0/0 form), OR both limits are ±∞ (∞/∞ form):

$$lim(x→a)$ f(x)/g(x) = $lim(x→a)$ f'(x)/g'(x)
$

Provided the limit on the right exists (or is ±∞).

Key: Differentiate NUMERATOR and DENOMINATOR SEPARATELY. Not the quotient rule!

Example: $lim(x→0)$ sin(x)/x 0/0 form. Apply L'Hôpital: = $lim(x→0)$ cos(x)/1 = 1/1 = 1

Example: $lim(x→∞)$ x/eˣ ∞/∞ form. Apply L'Hôpital: = $lim(x→∞)$ 1/eˣ = 0

Repeated Applications

Sometimes one application still gives 0/0 or ∞/∞.

Example: $lim(x→0)$ (sin(x) - x)/x³ First: (cos(x) - 1)/(3x²) → 0/0 Second: (-sin(x))/(6x) → 0/0 Third: (-cos(x))/6 → -1/6

Other Indeterminate Forms

0·∞

Rewrite as a fraction: 0·∞ = 0/(1/∞) or ∞/(1/0)

Example: $lim(x→0⁺)$ x·ln(x) = $lim(x→0⁺)$ ln(x)/(1/x) = (-∞)/(∞) Apply L'Hôpital: (1/x)/(-1/x²) = -x → 0

∞ - ∞

Combine into a single fraction.

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

Exponential forms (0⁰, 1^∞, ∞⁰)

Take natural log first.

Example: $lim(x→0⁺)$ xˣ Let y = xˣ, so ln(y) = x·ln(x) $lim(x→0⁺)$ x·ln(x) = $lim(x→0⁺)$ ln(x)/(1/x) = (-∞)/(∞) L'Hôpital: (1/x)/(-1/x²) = -x → 0 So ln(y) → 0, meaning y → e⁰ = 1

Key Terms

• 04 09 Lhopitals Rule
• Correct: A)
• Correct: B)
• Correct: C)
• Example 1: Basic L'Hôpital
• Example 2: Repeated
• Example 3: ∞ - ∞
• Exponential forms (0⁰, 1^∞, ∞⁰)
• Indeterminate Forms
• L'Hôpital's Rule
• Other Indeterminate Forms
• Repeated Applications

Worked Examples

Example 1: Basic L'Hôpital

$lim(x→0)$ (eˣ - 1)/x 0/0 form. L'Hôpital: eˣ/1 → 1/1 = 1

Example 2: Repeated

$lim(x→0)$ (1 - cos(x))/x² 0/0. First: sin(x)/(2x) → 0/0. Second: cos(x)/2 → 1/2

Example 3: ∞ - ∞

$lim(x→0)$ (1/x - 1/sin(x)) Direct: ∞ - ∞.

Combine: = $lim(x→0)$ (sin(x) - x)/(x·sin(x)) → 0/0 L'Hôpital: $lim(x→0)$ (cos(x) - 1)/(sin(x) + x·cos(x)) → 0/0 Second: $lim(x→0)$ (-sin(x))/(cos(x) + cos(x) - x·sin(x)) = $lim(x→0)$ (-sin(x))/(2cos(x) - x·sin(x)) = 0/(2) = 0

Quiz

Q1: What does the concept of Indeterminate Forms primarily refer to in this subject?

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

Correct: C)

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

Q2: What is the primary purpose of Other Indeterminate Forms?

A) It is used only in advanced research contexts B) It replaces all other methods in this domain C) It is used to other indeterminate forms in mathematical analysis D) It is primarily a historical notation system

Correct: C)

• If you chose A: This is incorrect. Other Indeterminate Forms serves the purpose described in the correct answer. The other options misrepresent its role.
• If you chose B: This is incorrect. Other Indeterminate Forms serves the purpose described in the correct answer. The other options misrepresent its role.
• If you chose C: Other Indeterminate Forms serves the purpose described in the correct answer. The other options misrepresent its role. Correct!
• If you chose D: This is incorrect. Other Indeterminate Forms serves the purpose described in the correct answer. The other options misrepresent its role.

Q3: Which statement about Repeated Applications is TRUE?

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

Correct: B)

• If you chose A: This is incorrect. Repeated Applications is a fundamental concept covered in this subject. This subject covers Repeated Applications as part of its core content.
• If you chose B: Repeated Applications is a fundamental concept covered in this subject. This subject covers Repeated Applications as part of its core content. Correct!
• If you chose C: This is incorrect. Repeated Applications is a fundamental concept covered in this subject. This subject covers Repeated Applications as part of its core content.
• If you chose D: This is incorrect. Repeated Applications is a fundamental concept covered in this subject. This subject covers Repeated Applications 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) An unrelated numerical value C) A different result from a common mistake D) 2.

Correct: D)

• If you chose A: This is incorrect. The worked examples show that the result is 2.. The other options represent common errors.
• If you chose B: This is incorrect. The worked examples show that the result is 2.. The other options represent common errors.
• If you chose C: This is incorrect. The worked examples show that the result is 2.. The other options represent common errors.
• If you chose D: The worked examples show that the result is 2.. The other options represent common errors. Correct!

Q5: How are Repeated Applications and L'Hôpital'S Rule related?

A) Repeated Applications and L'Hôpital'S Rule are completely unrelated topics B) Repeated Applications and L'Hôpital'S Rule are closely related concepts C) Repeated Applications is a special case of L'Hôpital'S Rule D) Repeated Applications is the inverse of L'Hôpital'S Rule

Correct: B)

• If you chose A: This is incorrect. Both Repeated Applications and L'Hôpital'S Rule are covered in this subject as interconnected topics.
• If you chose B: Both Repeated Applications and L'Hôpital'S Rule are covered in this subject as interconnected topics. Correct!
• If you chose C: This is incorrect. Both Repeated Applications and L'Hôpital'S Rule are covered in this subject as interconnected topics.
• If you chose D: This is incorrect. Both Repeated Applications and L'Hôpital'S Rule are covered in this subject as interconnected topics.

Q6: What is a common pitfall when working with ∞ - ∞?

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

Correct: C)

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

Q7: When should you apply Exponential Forms (0⁰, 1^∞, ∞⁰)?

A) Avoid Exponential Forms (0⁰, 1^∞, ∞⁰) unless explicitly instructed B) Use Exponential Forms (0⁰, 1^∞, ∞⁰) only in pure mathematics contexts C) Exponential Forms (0⁰, 1^∞, ∞⁰) is not practically useful D) Apply Exponential Forms (0⁰, 1^∞, ∞⁰) to solve problems in this subject's domain

Correct: D)

• If you chose A: This is incorrect. Exponential Forms (0⁰, 1^∞, ∞⁰) is a practical tool used throughout this subject to solve relevant problems.
• If you chose B: This is incorrect. Exponential Forms (0⁰, 1^∞, ∞⁰) is a practical tool used throughout this subject to solve relevant problems.
• If you chose C: This is incorrect. Exponential Forms (0⁰, 1^∞, ∞⁰) is a practical tool used throughout this subject to solve relevant problems.
• If you chose D: Exponential Forms (0⁰, 1^∞, ∞⁰) is a practical tool used throughout this subject to solve relevant problems. Correct!

Practice Problems

1. $lim(x→0)$ sin(2x)/x Answer: L'Hôpital: 2cos(2x)/1 → 2. Or use sin(2x) ≈ 2x, so limit = 2.

2. $lim(x→∞)$ ln(x)/x Answer: L'Hôpital: (1/x)/1 = 0.

3. $lim(x→0⁺)$ x·ln(x) Answer: Rewrite as ln(x)/(1/x). L'Hôpital: (1/x)/(-1/x²) = -x → 0.

4. $lim(x→0)$ (eˣ - 1 - x)/x² Answer: 0/0. First: (eˣ - 1)/(2x) → 0/0. Second: eˣ/2 → 1/2.

5. $lim(x→∞)$ (1 + 1/x)ˣ Answer: Let y = (1 + 1/x)ˣ. ln(y) = x·ln(1 + 1/x) = ln(1 + 1/x)/(1/x). L'Hôpital: (1/(1+1/x)·(-1/x²))/(-1/x²) = 1/(1+1/x) → 1. So y → e¹ = e.

Summary

Key takeaways:

• L'Hôpital: for 0/0 or ∞/∞, differentiate num and denom separately
• May need multiple applications
• Convert other forms (0·∞, ∞-∞, exponentials) to 0/0 or ∞/∞ first
• For 1^∞, 0⁰, ∞⁰: take ln, apply L'Hôpital, then exponentiate

Pitfalls

• Applying L'Hôpital to non-indeterminate forms. L'Hôpital's Rule only works for 0/0 or ∞/∞. Applying it to something like $lim(x→0)$ x/2 (which is simply 0/2 = 0) gives the wrong answer. Always check the form first.
• Using the quotient rule instead of differentiating separately. L'Hôpital says differentiate numerator and denominator independently: f'(x)/g'(x), NOT (f'g - fg')/g². The quotient rule does something entirely different.
• Stopping after one application when the result is still indeterminate. After one round, if you still have 0/0 or ∞/∞, apply L'Hôpital again. Limits like (sin x - x)/x³ need three applications.
• Not converting other indeterminate forms. 0·∞, ∞-∞, 0⁰, 1^∞, and ∞⁰ all require algebraic manipulation before L'Hôpital applies. You cannot directly differentiate these forms.
• Assuming L'Hôpital always works. If lim f'(x)/g'(x) does not exist (oscillates), L'Hôpital gives no information — the original limit may still exist by other means.

Next Steps

Next up: 04-10-newtons-method.md