Unit 1.6 – Determining Limits Using Algebraic Manipulation
AP® Calculus AB & BC | Formula Reference Sheet
Core Concept: When direct substitution gives an indeterminate form (like 0/0), we need algebraic manipulation to rewrite the function into an equivalent form that can be evaluated. This unit teaches you the essential "rescue techniques" for evaluating tricky limits!
🎯 When Do You Need Algebraic Manipulation?
Step 1: Try direct substitution first (plug in the value)
- If you get a finite number: Done! That's your limit ✓
- If you get 0/0: Use algebraic manipulation (this unit!) 🔧
- If you get nonzero/0: Infinite limit or DNE (vertical asymptote)
- If you get ∞/∞: Use algebraic manipulation (divide by highest power)
❓ Indeterminate Forms
The 7 Indeterminate Forms:
Most common in AP® Calculus: \(\frac{0}{0}\) and \(\frac{\infty}{\infty}\)
These forms are "indeterminate" because they could equal ANY value—you need more information (algebraic manipulation) to find the actual limit.
📝 Key Point: When you get 0/0, it means there's a removable discontinuity (hole) at that point. Algebraic manipulation helps you "see" what the function approaches near the hole.
🔧 Technique 1: Factoring and Canceling
Goal: Factor numerator and denominator, then cancel common factors that create 0/0
Steps:
- Try direct substitution → Get 0/0? Proceed
- Factor numerator completely
- Factor denominator completely
- Cancel common factors (these cause the 0/0)
- Substitute again into simplified expression
Example 1: Simple Factoring
Find: \(\lim_{x \to 2} \frac{x^2 - 4}{x - 2}\)
Solution:
- Direct substitution: \(\frac{0}{0}\) ← indeterminate!
- Factor numerator: \(x^2 - 4 = (x-2)(x+2)\)
- Rewrite: \(\frac{(x-2)(x+2)}{x-2}\)
- Cancel \((x-2)\): \(x + 2\) (valid for \(x \neq 2\))
- Substitute: \(\lim_{x \to 2} (x+2) = 4\)
Answer: 4
Example 2: Quadratic Factoring
Find: \(\lim_{x \to 3} \frac{x^2 - 5x + 6}{x^2 - 9}\)
Solution:
- Direct substitution: \(\frac{0}{0}\)
- Factor numerator: \((x-2)(x-3)\)
- Factor denominator: \((x-3)(x+3)\)
- Cancel \((x-3)\): \(\frac{x-2}{x+3}\)
- Substitute: \(\frac{3-2}{3+3} = \frac{1}{6}\)
Answer: \(\frac{1}{6}\)
🌟 Technique 2: Rationalizing (Conjugate Method)
Goal: Multiply by the conjugate to eliminate radicals and create factorable expressions
The conjugate of \(a + b\) is \(a - b\) (flip the middle sign)
Key Identity: \((a+b)(a-b) = a^2 - b^2\) ← eliminates radicals!
Steps:
- Identify which part has the radical (numerator or denominator)
- Multiply top AND bottom by the conjugate of that part
- Expand using \((a+b)(a-b) = a^2 - b^2\)
- Simplify and cancel common factors
- Substitute the limit value
Example 3: Radical in Numerator
Find: \(\lim_{x \to 9} \frac{\sqrt{x} - 3}{x - 9}\)
Solution:
- Direct substitution: \(\frac{0}{0}\)
- Conjugate: \(\sqrt{x} + 3\)
- Multiply: \(\frac{\sqrt{x} - 3}{x - 9} \cdot \frac{\sqrt{x} + 3}{\sqrt{x} + 3}\)
- Numerator: \((\sqrt{x})^2 - 3^2 = x - 9\)
- Result: \(\frac{x-9}{(x-9)(\sqrt{x}+3)}\)
- Cancel \((x-9)\): \(\frac{1}{\sqrt{x}+3}\)
- Substitute: \(\frac{1}{\sqrt{9}+3} = \frac{1}{6}\)
Answer: \(\frac{1}{6}\)
Example 4: Radical in Denominator
Find: \(\lim_{x \to 4} \frac{x - 4}{\sqrt{x} - 2}\)
Solution:
- Direct substitution: \(\frac{0}{0}\)
- Conjugate of denominator: \(\sqrt{x} + 2\)
- Multiply: \(\frac{x - 4}{\sqrt{x} - 2} \cdot \frac{\sqrt{x} + 2}{\sqrt{x} + 2}\)
- Denominator: \(x - 4\)
- Result: \(\frac{(x-4)(\sqrt{x}+2)}{x-4}\)
- Cancel \((x-4)\): \(\sqrt{x} + 2\)
- Substitute: \(\sqrt{4} + 2 = 4\)
Answer: 4
Example 5: Both Have Radicals
Find: \(\lim_{x \to 0} \frac{\sqrt{1+x} - 1}{x}\)
Solution:
- Direct substitution: \(\frac{0}{0}\)
- Conjugate: \(\sqrt{1+x} + 1\)
- Multiply: \(\frac{\sqrt{1+x} - 1}{x} \cdot \frac{\sqrt{1+x} + 1}{\sqrt{1+x} + 1}\)
- Numerator: \((1+x) - 1 = x\)
- Result: \(\frac{x}{x(\sqrt{1+x}+1)}\)
- Cancel \(x\): \(\frac{1}{\sqrt{1+x}+1}\)
- Substitute: \(\frac{1}{\sqrt{1}+1} = \frac{1}{2}\)
Answer: \(\frac{1}{2}\)
➗ Technique 3: Common Denominator (Complex Fractions)
Goal: Combine fractions into a single fraction, then simplify
Example 6: Complex Fraction
Find: \(\lim_{x \to 3} \frac{\frac{1}{x} - \frac{1}{3}}{x - 3}\)
Solution:
- Direct substitution: \(\frac{0}{0}\)
- Common denominator in numerator: \(\frac{3 - x}{3x}\)
- Rewrite: \(\frac{\frac{3-x}{3x}}{x-3}\)
- Division of fractions: \(\frac{3-x}{3x} \cdot \frac{1}{x-3}\)
- Factor: \(\frac{-(x-3)}{3x(x-3)}\)
- Cancel \((x-3)\): \(\frac{-1}{3x}\)
- Substitute: \(\frac{-1}{3(3)} = -\frac{1}{9}\)
Answer: \(-\frac{1}{9}\)
📐 Technique 4: Trigonometric Limits
The "Big Three" you MUST memorize:
📝 Variations: For any constant \(k\):
Example 7: Basic Trig Limit
Find: \(\lim_{x \to 0} \frac{\sin(5x)}{x}\)
Solution:
- Goal: Get \(\frac{\sin(5x)}{5x}\) form (which equals 1)
- Multiply by \(\frac{5}{5}\): \(\frac{\sin(5x)}{x} \cdot \frac{5}{5} = 5 \cdot \frac{\sin(5x)}{5x}\)
- Apply formula: \(5 \cdot 1 = 5\)
Answer: 5
Example 8: Trig with Algebra
Find: \(\lim_{x \to 0} \frac{\sin(3x)}{2x}\)
Solution:
- Rewrite: \(\frac{3}{2} \cdot \frac{\sin(3x)}{3x}\)
- Apply formula: \(\frac{3}{2} \cdot 1 = \frac{3}{2}\)
Answer: \(\frac{3}{2}\)
♾️ Technique 5: Limits at Infinity (Rational Functions)
Goal: Divide numerator and denominator by highest power of \(x\)
For \(\lim_{x \to \infty} \frac{p(x)}{q(x)}\) where \(p\) and \(q\) are polynomials:
- Degree of numerator < denominator: Limit = 0
- Degree of numerator = denominator: Limit = ratio of leading coefficients
- Degree of numerator > denominator: Limit = ±∞
Example 9: Limit at Infinity
Find: \(\lim_{x \to \infty} \frac{3x^2 + 5x - 1}{7x^2 - 2x + 4}\)
Solution:
- Same degree (both \(x^2\))
- Leading coefficients: 3 and 7
- Answer: \(\frac{3}{7}\)
Alternative method (divide by \(x^2\)):
📋 Quick Reference: Which Technique to Use
| If You See... | Use This Technique | Example |
|---|---|---|
| Polynomial ÷ Polynomial (0/0) | Factor and Cancel | \(\frac{x^2-4}{x-2}\) |
| Square root in numerator | Multiply by conjugate | \(\frac{\sqrt{x}-2}{x-4}\) |
| Square root in denominator | Multiply by conjugate | \(\frac{x-9}{\sqrt{x}-3}\) |
| Fraction ÷ Fraction | Common denominator | \(\frac{\frac{1}{x}-\frac{1}{a}}{x-a}\) |
| \(\frac{\sin(\text{stuff})}{\text{stuff}}\) | Trig limit formula | \(\frac{\sin(3x)}{x}\) |
| \(x \to \infty\) with rational function | Divide by highest power | \(\frac{2x^2+1}{3x^2-5}\) |
| Absolute value | One-sided limits | \(\frac{|x-2|}{x-2}\) |
📏 Special Case: Absolute Value Limits
When you have \(|x-a|\) in the expression and \(x \to a\), evaluate one-sided limits:
- From left (\(x < a\)): \(|x-a| = -(x-a) = a-x\)
- From right (\(x > a\)): \(|x-a| = x-a\)
Example 10: Absolute Value
Find: \(\lim_{x \to 2} \frac{|x-2|}{x-2}\)
Solution:
- Left-hand limit (\(x < 2\)): \(\frac{-(x-2)}{x-2} = -1\)
- Right-hand limit (\(x > 2\)): \(\frac{x-2}{x-2} = 1\)
- Since \(-1 \neq 1\): Limit DNE
💡 Tips, Tricks & Strategies
✅ Essential Tips
- ALWAYS try direct substitution first! Don't waste time manipulating if it's not needed
- Factor everything you can: Look for GCF, difference of squares, quadratics
- Conjugates work miracles: When you see radicals and get 0/0, think conjugate!
- Match the form: For trig limits, manipulate to get \(\frac{\sin(\text{thing})}{\text{thing}}\)
- Show your work: On AP® exams, you need to justify algebraic steps
- Check your cancellation: Make sure you're not dividing by zero
🎯 The "Algebraic Manipulation Checklist"
When you get 0/0, ask yourself:
- Can I factor? → Try factoring numerator and denominator
- Is there a radical? → Try multiplying by the conjugate
- Is it a complex fraction? → Find common denominator
- Is it trig? → Use trig identities and special limits
- Is it at infinity? → Divide by highest power
🔥 Common Factoring Patterns
| Pattern | Factored Form |
|---|---|
| \(a^2 - b^2\) | \((a-b)(a+b)\) |
| \(x^2 + bx + c\) | \((x+p)(x+q)\) where \(p+q=b, pq=c\) |
| \(ax^2 + bx + c\) | Use AC method or quadratic formula |
| \(a^3 - b^3\) | \((a-b)(a^2+ab+b^2)\) |
| \(a^3 + b^3\) | \((a+b)(a^2-ab+b^2)\) |
❌ Common Mistakes to Avoid
- Mistake 1: Canceling terms that aren't factors (you can only cancel factors, not terms!)
- Mistake 2: Forgetting to multiply BOTH numerator and denominator by conjugate
- Mistake 3: Stopping at 0/0 without doing manipulation
- Mistake 4: Canceling before checking if the canceled factor makes x = a undefined
- Mistake 5: Not simplifying completely before substituting
- Mistake 6: Forgetting to check one-sided limits for absolute values
- Mistake 7: Using wrong conjugate (\(a+b\) and \(a+b\) won't work!)
📚 Summary of Techniques with Examples
| Technique | Example Problem | Key Step |
|---|---|---|
| Factoring | \(\lim_{x \to 2} \frac{x^2-4}{x-2}\) | Factor: \(\frac{(x-2)(x+2)}{x-2} = x+2\) |
| Conjugate | \(\lim_{x \to 4} \frac{\sqrt{x}-2}{x-4}\) | Multiply by \(\frac{\sqrt{x}+2}{\sqrt{x}+2}\) |
| Common Denom. | \(\lim_{x \to 3} \frac{\frac{1}{x}-\frac{1}{3}}{x-3}\) | Combine: \(\frac{3-x}{3x(x-3)}\) |
| Trig Limits | \(\lim_{x \to 0} \frac{\sin(5x)}{x}\) | Rewrite: \(5 \cdot \frac{\sin(5x)}{5x}\) |
| At Infinity | \(\lim_{x \to \infty} \frac{3x^2+1}{5x^2-2}\) | Leading coefficients: \(\frac{3}{5}\) |
✏️ AP® Exam Tips
What the AP® Exam Expects:
- Show ALL algebraic steps: Don't just write the answer—show factoring, conjugate multiplication, etc.
- Justify cancellations: Write "for \(x \neq a\)" when canceling factors
- State the form: Write "This gives 0/0, so we need to manipulate..."
- Know when to stop: After canceling, state the simplified form before substituting
- Use proper notation: Write \(\lim_{x \to a}\), not just "limit"
- Check for DNE: If one-sided limits differ, explicitly state "DNE"
- Calculator note: On non-calculator sections, you MUST show algebraic work
- Time management: Don't get stuck—if one technique doesn't work, try another!
📝 How to Master Algebraic Manipulation
Effective Practice Strategy:
- Master factoring first: Review all factoring patterns (it's the foundation!)
- Practice conjugate multiplication: Do 10+ examples until it's automatic
- Memorize trig limits: Flash cards for the "Big Three"
- Identify the type BEFORE solving: Look at the problem and name the technique you'll use
- Work backwards: Given a simplified form, create a 0/0 problem that requires manipulation
- Time yourself: Speed matters on the AP® exam
- Redo mistakes: Keep a "mistake log" of errors you've made
⚡ Quick Reference Decision Tree
Got 0/0? Follow this:
1. Is it polynomial/polynomial? → Factor and cancel
2. Do you see \(\sqrt{\text{anything}}\)? → Multiply by conjugate
3. Is it fraction/fraction? → Find common denominator
4. Do you see \(\sin(\text{stuff})\)? → Use trig limit formulas
5. Is \(x \to \infty\)? → Divide by highest power
6. Still stuck? → Check for absolute values, one-sided limits, or special identities
🔗 Connection to Other Topics
Unit 1.6 is critical for:
- Unit 1.7-1.16: Continuity and types of discontinuities (removable = 0/0 that you can fix!)
- Unit 2.1: Definition of derivative (uses \(\frac{0}{0}\) limits)
- Unit 2.4: Derivatives of trig functions (uses trig limits)
- Unit 3.4: L'Hôpital's Rule (advanced tool for indeterminate forms)
- Unit 6: Improper integrals (limits at infinity)
- Throughout calculus: Algebraic manipulation is EVERYWHERE!
Remember: When direct substitution gives 0/0, it's not a dead end—it's an invitation to use algebraic manipulation! Master these five techniques (factoring, conjugate, common denominator, trig formulas, and division by highest power), and you'll be able to handle almost any limit problem the AP® Calculus exam throws at you. Practice until these techniques become second nature! 🎯🔧✨