Unit 1.7 – Selecting Procedures for Determining Limits
AP® Calculus AB & BC | Formula Reference Sheet
Core Concept: This unit synthesizes everything from Units 1.2-1.6 into a strategic decision-making framework. You now have multiple tools for finding limits—the key is knowing which tool to use when. Think of this as your "limit problem triage system"!
🌟 The Golden Rule of Limit Evaluation
Direct Substitution: Plug in the value and see what happens
Why this first? If it works, you're done in seconds! Don't waste time with complex manipulation if direct substitution gives you a finite answer.
⚠️ Critical Point: Use other methods only when direct substitution fails. If you get a real number from direct substitution, that's your answer—don't overthink it!
🎯 The Master Decision Flowchart
STEP-BY-STEP LIMIT EVALUATION FLOWCHART
STEP 1: What format is the problem in?
- Given a GRAPH? → Use visual/graphical method (Unit 1.3)
- Given a TABLE? → Use numerical method (Unit 1.4)
- Given an EXPRESSION? → Proceed to Step 2
STEP 2: Try DIRECT SUBSTITUTION
Plug \(x = a\) into the expression. What do you get?
- Finite number? → DONE! That's your limit ✓
- \(\frac{0}{0}\)? → Proceed to Step 3 (indeterminate form)
- \(\frac{\text{nonzero}}{0}\)? → Infinite limit or DNE (vertical asymptote)
- \(\frac{\infty}{\infty}\)? → Proceed to Step 4 (limits at infinity)
STEP 3: Got \(\frac{0}{0}\)? Choose algebraic manipulation:
- Polynomial/rational? → Factor and cancel
- Square roots (\(\sqrt{\text{stuff}}\))? → Multiply by conjugate
- Complex fraction? → Find common denominator
- Trig functions? → Use trig limit theorems
- Piecewise? → Check one-sided limits
STEP 4: Limits at Infinity (\(x \to \pm\infty\))?
- Rational function? → Divide by highest power OR compare degrees
- Exponential? → Consider dominant term (exponentials dominate polynomials)
STEP 5: Special cases:
- Absolute values? → Check one-sided limits
- Oscillating (like \(\sin(1/x)\))? → Squeeze Theorem
📚 Quick Review: All Available Methods
Method 1: Graphical Estimation (Unit 1.3)
When to use: Problem gives you a graph
How: Trace the curve from both sides toward \(x = a\); observe what y-value is approached
Pros: Quick visual understanding
Cons: Limited by scale, only gives estimates
Method 2: Numerical Estimation (Unit 1.4)
When to use: Problem gives you a table of values
How: Look at function values as x-values approach target from both sides
Pros: Clear convergence patterns
Cons: Can miss behavior between table entries
Method 3: Direct Substitution (Unit 1.5)
When to use: ALWAYS TRY FIRST! When function is continuous at \(x = a\)
How: Simply plug in: \(\lim_{x \to a} f(x) = f(a)\)
Works for: Polynomials, most rational functions, trig functions (at valid points)
Method 4: Algebraic Properties (Unit 1.5)
When to use: Breaking complex expressions into simpler pieces
Tools: Sum/difference, product, quotient, power, composition laws
Example: \(\lim_{x \to 2} [3x^2 + \sin(x)] = 3\lim_{x \to 2} x^2 + \lim_{x \to 2} \sin(x)\)
Method 5: Algebraic Manipulation (Unit 1.6)
When to use: When direct substitution gives \(\frac{0}{0}\)
Techniques:
- Factoring: For polynomials
- Conjugate: For radicals
- Common denominator: For complex fractions
- Trig identities: For trig expressions
- Divide by highest power: For limits at infinity
🔍 Quick Decision Matrix
| What You See | What You Try | Why |
|---|---|---|
| A graph is given | Visual method | Direct observation of behavior |
| A table of (x,y) values | Numerical method | Look for convergence patterns |
| Any expression | Direct substitution FIRST | Fastest method if it works |
| \(\frac{\text{polynomial}}{\text{polynomial}}\) giving 0/0 | Factor and cancel | Removes common factors |
| \(\frac{\sqrt{\text{stuff}}}{\text{anything}}\) giving 0/0 | Multiply by conjugate | Eliminates radical |
| \(\frac{\sin(kx)}{x}\) as \(x \to 0\) | Trig limit theorem | Use \(\frac{\sin u}{u} \to 1\) |
| \(x \to \infty\) with rational function | Divide by highest power | Find horizontal asymptote |
| Piecewise function at boundary | One-sided limits | Check left and right separately |
| Absolute value \(|x-a|\) at \(x = a\) | One-sided limits | Definition changes sign |
📖 Worked Examples: Choosing the Right Method
Example 1: Direct Substitution Works
Find: \(\lim_{x \to 3} (2x^2 + 5x - 1)\)
Step 1: Try direct substitution
Result: \(2(3)^2 + 5(3) - 1 = 18 + 15 - 1 = 32\)
Decision: ✓ Direct substitution worked! Answer = 32
Method Used: Direct Substitution
Example 2: Needs Factoring
Find: \(\lim_{x \to 2} \frac{x^2 - 4}{x - 2}\)
Step 1: Try direct substitution → \(\frac{0}{0}\) ✗
Step 2: Got 0/0, so try algebraic manipulation
Step 3: See polynomial over polynomial → Factor!
Work: \(\frac{(x-2)(x+2)}{x-2} = x+2\) for \(x \neq 2\)
Step 4: Now substitute: \(2 + 2 = 4\)
Method Used: Algebraic Manipulation (Factoring)
Example 3: Needs Conjugate
Find: \(\lim_{x \to 0} \frac{\sqrt{x+1} - 1}{x}\)
Step 1: Try direct substitution → \(\frac{0}{0}\) ✗
Step 2: See square root → Use conjugate!
Work: Multiply by \(\frac{\sqrt{x+1}+1}{\sqrt{x+1}+1}\)
Result: \(\frac{x}{x(\sqrt{x+1}+1)} = \frac{1}{\sqrt{x+1}+1}\)
Step 3: Substitute: \(\frac{1}{2}\)
Method Used: Algebraic Manipulation (Conjugate)
Example 4: Needs Trig Theorem
Find: \(\lim_{x \to 0} \frac{\sin(3x)}{x}\)
Step 1: Try direct substitution → \(\frac{0}{0}\) ✗
Step 2: See \(\frac{\sin(\text{stuff})}{x}\) → Use trig theorem!
Work: \(\frac{\sin(3x)}{x} = 3 \cdot \frac{\sin(3x)}{3x}\)
Apply: \(\lim_{u \to 0} \frac{\sin u}{u} = 1\)
Result: \(3 \cdot 1 = 3\)
Method Used: Trig Limit Theorem
Example 5: Limit at Infinity
Find: \(\lim_{x \to \infty} \frac{3x^2 + 5}{2x^2 - 7}\)
Step 1: See \(x \to \infty\) → Not direct substitution
Step 2: Rational function at infinity → Compare degrees
Analysis: Same degree (both \(x^2\))
Result: Ratio of leading coefficients = \(\frac{3}{2}\)
Method Used: Limits at Infinity (Degree Comparison)
Example 6: One-Sided Limits Needed
Find: \(\lim_{x \to 2} f(x)\) where \(f(x) = \begin{cases} x^2 & x < 2 \\ 2x & x \geq 2 \end{cases}\)
Step 1: Piecewise function → Check one-sided limits!
Left: \(\lim_{x \to 2^-} x^2 = 4\)
Right: \(\lim_{x \to 2^+} 2x = 4\)
Compare: Both equal 4 → Limit = 4
Method Used: One-Sided Limits
🎓 "When to Use What" Summary
Use GRAPHICAL METHOD when:
- Problem explicitly provides a graph
- Asked for estimation/approximation
- Quick visual check needed
Use NUMERICAL METHOD when:
- Problem provides a table of values
- Verifying algebraic work
- Calculator table mode appropriate
Use DIRECT SUBSTITUTION when:
- ALWAYS TRY THIS FIRST
- Function is continuous at the point
- Polynomials, simple rational functions
- You get a finite answer
Use FACTORING when:
- Direct substitution gives \(\frac{0}{0}\)
- Numerator and denominator are polynomials
- You can find common factors
Use CONJUGATE when:
- Direct substitution gives \(\frac{0}{0}\)
- Square roots (\(\sqrt{}\)) in numerator OR denominator
- Expression like \(\sqrt{x+h} - \sqrt{x}\)
Use TRIG THEOREMS when:
- \(\sin(x)\), \(\cos(x)\), or \(\tan(x)\) appear
- Approaching 0 or multiples of \(\pi\)
- Form looks like \(\frac{\sin(\text{stuff})}{\text{stuff}}\)
Use ONE-SIDED LIMITS when:
- Piecewise function at boundary point
- Absolute value at the critical point
- Checking continuity
- Left and right behavior might differ
❌ Common Mistakes in Method Selection
- Mistake 1: Not trying direct substitution first — Always start here! Don't waste time factoring if it's unnecessary
- Mistake 2: Trying to factor when you see radicals — Use conjugate instead!
- Mistake 3: Using graphical estimates when exact answer is required
- Mistake 4: Ignoring 0/0 and stopping — This means use algebraic manipulation!
- Mistake 5: Forgetting to check one-sided limits for piecewise functions
- Mistake 6: Using wrong technique for limits at infinity (don't factor—divide by highest power!)
- Mistake 7: Not recognizing trig limit patterns
- Mistake 8: Giving up too quickly — Try multiple techniques if needed
💡 Master Strategies & Tips
✅ Essential Tips
- Direct substitution is ALWAYS step 1 — Never skip this!
- 0/0 means "keep going" — It's not an answer, it's a signal to manipulate
- Match patterns — Learn to recognize "factoring problems" vs "conjugate problems"
- Check both sides — For piecewise, absolute values, always verify left and right
- Know your tools — Memorize when each technique applies
- Work efficiently — Use the simplest method that works
🎯 The "Problem Diagnosis" Skill
Before you start solving, ask yourself:
- What format? Graph, table, or expression?
- What type of function? Polynomial, rational, radical, trig?
- Where's it going? Finite value, infinity, or boundary point?
- What happens with direct substitution? Finite, 0/0, or undefined?
- What technique fits? Match to decision matrix
This diagnosis takes 5-10 seconds and saves minutes of wasted work!
🔥 The "Backup Plan" Strategy
If your first technique doesn't work:
- Step back and reassess — Did you choose the right method?
- Try a different manipulation — Sometimes multiple approaches work
- Check for special patterns — Trig identities, difference of squares, etc.
- Use numerical verification — Make a quick table to check your answer
📝 Practice: Identify the Method
For each limit, identify which method(s) you would use:
1. \(\lim_{x \to 5} (3x^2 - 2x + 1)\)
Answer: Direct substitution (polynomial, continuous everywhere)
2. \(\lim_{x \to 3} \frac{x^2 - 9}{x - 3}\)
Answer: Direct substitution gives 0/0 → Factor and cancel
3. \(\lim_{x \to 4} \frac{\sqrt{x} - 2}{x - 4}\)
Answer: Direct substitution gives 0/0 → Multiply by conjugate
4. \(\lim_{x \to 0} \frac{\sin(5x)}{x}\)
Answer: Direct substitution gives 0/0 → Use trig limit theorem
5. \(\lim_{x \to \infty} \frac{2x^3 + 1}{5x^3 - 7}\)
Answer: Limit at infinity → Compare degrees (answer: 2/5)
6. Given a graph of \(f(x)\), find \(\lim_{x \to 2} f(x)\)
Answer: Graphical method → Trace curve from both sides
✏️ AP® Exam Success Strategies
What the AP® Exam Tests:
- Method selection — Can you choose the right tool?
- Justification — Can you explain why you chose that method?
- Execution — Can you carry out the technique correctly?
- Verification — Can you check if your answer makes sense?
FRQ Expectations:
- Show your initial attempt — Write "Try direct substitution: \(f(a) = ...\)"
- State the indeterminate form — Write "This gives 0/0" if applicable
- Name your technique — "Factor and cancel" or "Multiply by conjugate"
- Show all algebraic steps — Don't skip work!
- State the final answer clearly — Box or underline it
Multiple Choice Strategies:
- Time management: Direct substitution should be instant—don't overthink
- Process of elimination: Rule out answers that violate limit properties
- Check reasonableness: Does the answer match the function's behavior?
- Calculator check: When allowed, verify with table or graph
⚡ Quick Reference Card
| Step | What to Do | Decision Point |
|---|---|---|
| 1 | Check problem format | Graph? → Visual | Table? → Numerical | Expression? → Continue |
| 2 | Try direct substitution | Finite? → DONE | 0/0? → Step 3 | nonzero/0? → Infinite |
| 3 | Identify structure | Polynomial? → Factor | Radical? → Conjugate | Trig? → Theorem |
| 4 | Apply technique | Manipulate algebraically to eliminate 0/0 |
| 5 | Substitute again | Get finite answer from simplified form |
| 6 | Verify | Does answer make sense? Check with table/graph if unsure |
🔗 Why This Unit Matters
Unit 1.7 is the synthesis of all limit techniques—mastering method selection prepares you for:
- Unit 1.8-1.16: Continuity analysis (need limits to test continuity)
- Unit 2: Derivatives (defined using limits—you'll need manipulation skills)
- Unit 3: Analyzing function behavior (limits describe end behavior)
- Unit 6: Improper integrals (evaluate using limits)
- AP® Exam success: Fast, accurate method selection is crucial under time pressure
- Problem-solving mindset: Learning to diagnose problems is a life skill!
Remember: Selecting the right procedure is like being a doctor diagnosing a patient—you need to recognize the symptoms (problem type), understand the tools (methods), and apply the right treatment (technique). ALWAYS start with direct substitution—it's your fastest tool. When that fails, match the problem structure to the technique: polynomials → factor, radicals → conjugate, trig → theorems, infinity → highest power. Master this skill, and you'll fly through limit problems with confidence! 🎯🚀