Unit 1.5 – Determining Limits Using Algebraic Properties of Limits

AP® Calculus AB & BC | Formula Reference Sheet

Core Concept: The algebraic properties of limits (also called limit laws or limit theorems) allow us to break down complex limit problems into simpler pieces. These properties are the foundation for evaluating limits without graphing or tables—just using algebra!

🎯 The Direct Substitution Principle

When Can You Just "Plug In"?

If \(f(x)\) is continuous at \(x = a\), then:

\[ \lim_{x \to a} f(x) = f(a) \]

Meaning: For continuous functions (polynomials, most trig functions, exponentials, etc.), you can find the limit by direct substitution—just plug in the value!

📝 Key Point: Direct substitution works for:

Polynomials: \(\lim_{x \to a} p(x) = p(a)\)
Rational functions: \(\lim_{x \to a} \frac{p(x)}{q(x)} = \frac{p(a)}{q(a)}\) (if \(q(a) \neq 0\))
Radicals: \(\lim_{x \to a} \sqrt[n]{f(x)} = \sqrt[n]{f(a)}\) (when defined)
Trig functions: \(\lim_{x \to a} \sin(x) = \sin(a)\), etc. (at valid points)

📚 The Complete Limit Laws

Given: \(\lim_{x \to a} f(x) = L\) and \(\lim_{x \to a} g(x) = M\) (both exist and are finite)

Then the following properties hold:

1. Constant Law

Limit of a Constant

\[ \lim_{x \to a} c = c \]

Rule: The limit of a constant is just the constant itself.

Example: \(\lim_{x \to 5} 7 = 7\)

2. Identity Law

Limit of x

\[ \lim_{x \to a} x = a \]

Rule: As \(x\) approaches \(a\), the limit of \(x\) is \(a\).

Example: \(\lim_{x \to 3} x = 3\)

3. Sum Law

Limit of a Sum

\[ \lim_{x \to a} [f(x) + g(x)] = L + M \]

Rule: The limit of a sum is the sum of the limits.

Example: \(\lim_{x \to 2} (x^2 + 3x) = \lim_{x \to 2} x^2 + \lim_{x \to 2} 3x = 4 + 6 = 10\)

4. Difference Law

Limit of a Difference

\[ \lim_{x \to a} [f(x) - g(x)] = L - M \]

Rule: The limit of a difference is the difference of the limits.

Example: \(\lim_{x \to 3} (x^2 - 2x) = 9 - 6 = 3\)

5. Constant Multiple Law

Limit of a Constant Times a Function

\[ \lim_{x \to a} [c \cdot f(x)] = c \cdot L \]

Rule: The limit of a constant times a function is the constant times the limit of the function.

Example: \(\lim_{x \to 4} 5x^2 = 5 \cdot \lim_{x \to 4} x^2 = 5 \cdot 16 = 80\)

6. Product Law

Limit of a Product

\[ \lim_{x \to a} [f(x) \cdot g(x)] = L \cdot M \]

Rule: The limit of a product is the product of the limits.

Example: \(\lim_{x \to 2} (x \cdot x^2) = 2 \cdot 4 = 8\)

7. Quotient Law

Limit of a Quotient

\[ \lim_{x \to a} \frac{f(x)}{g(x)} = \frac{L}{M} \quad \text{(provided } M \neq 0\text{)} \]

Rule: The limit of a quotient is the quotient of the limits, as long as the denominator's limit is not zero.

Example: \(\lim_{x \to 3} \frac{x^2 + 1}{2x} = \frac{10}{6} = \frac{5}{3}\)

⚠️ Critical Condition: The quotient law only works when \(M \neq 0\). If the denominator's limit is zero, you cannot use this property directly—you need algebraic manipulation (see Unit 1.6).

8. Power Law

Limit of a Power

\[ \lim_{x \to a} [f(x)]^n = L^n \]

Rule: The limit of a function to a power \(n\) is the limit raised to that power.

Example: \(\lim_{x \to 2} (x + 1)^3 = 3^3 = 27\)

9. Root Law

Limit of a Root

\[ \lim_{x \to a} \sqrt[n]{f(x)} = \sqrt[n]{L} \]

Conditions:

If \(n\) is odd: Works for all \(L\)
If \(n\) is even: \(L \geq 0\) required

Example: \(\lim_{x \to 4} \sqrt{2x + 1} = \sqrt{9} = 3\)

📋 Quick Reference: All Limit Laws

Property	Formula	Condition
Constant	\(\lim_{x \to a} c = c\)	Always
Identity	\(\lim_{x \to a} x = a\)	Always
Sum	\(\lim [f + g] = L + M\)	Both limits exist
Difference	\(\lim [f - g] = L - M\)	Both limits exist
Constant Multiple	\(\lim [c \cdot f] = c \cdot L\)	Limit exists
Product	\(\lim [f \cdot g] = L \cdot M\)	Both limits exist
Quotient	\(\lim \frac{f}{g} = \frac{L}{M}\)	Both exist, \(M \neq 0\)
Power	\(\lim [f]^n = L^n\)	Limit exists
Root	\(\lim \sqrt[n]{f} = \sqrt[n]{L}\)	\(L \geq 0\) for even \(n\)

🔗 Limits of Composite Functions

Composition Law (Continuity Required)

If \(\lim_{x \to a} g(x) = L\) and \(f\) is continuous at \(L\), then:

\[ \lim_{x \to a} f(g(x)) = f\left(\lim_{x \to a} g(x)\right) = f(L) \]

In words: For continuous outer functions, you can "pull the limit inside."

Example: \(\lim_{x \to 2} \sin(x^2)\)

Inner limit: \(\lim_{x \to 2} x^2 = 4\)
Outer function: \(\sin(x)\) is continuous at \(x = 4\)
Apply composition law: \(\lim_{x \to 2} \sin(x^2) = \sin(4)\)

🎯 How to Apply the Limit Laws

General Strategy:

Try direct substitution first: If the function is continuous, just plug in the value
Break into pieces: Use sum, difference, product, quotient laws to separate complex expressions
Evaluate each piece: Find the limit of each component separately
Combine results: Put the pieces back together using the appropriate law
Check conditions: Make sure denominators aren't zero, roots are defined, etc.

📚 Worked Examples

Example 1: Polynomial Limit

Find: \(\lim_{x \to 3} (2x^2 + 5x - 1)\)

Solution: Polynomials are continuous everywhere, so use direct substitution:

\[ \lim_{x \to 3} (2x^2 + 5x - 1) = 2(3)^2 + 5(3) - 1 = 18 + 15 - 1 = 32 \]

Example 2: Rational Function (Valid Denominator)

Find: \(\lim_{x \to 2} \frac{x^3 - 4x}{x + 3}\)

Solution:

Check denominator: \(\lim_{x \to 2} (x + 3) = 5 \neq 0\) ✓
Apply quotient law:

\[ \lim_{x \to 2} \frac{x^3 - 4x}{x + 3} = \frac{\lim_{x \to 2} (x^3 - 4x)}{\lim_{x \to 2} (x + 3)} = \frac{8 - 8}{5} = \frac{0}{5} = 0 \]

Example 3: Using Multiple Laws

Find: \(\lim_{x \to 1} 3(x^2 + 2)^4\)

Solution:

Constant multiple law: \(\lim 3(x^2 + 2)^4 = 3 \cdot \lim (x^2 + 2)^4\)
Power law: \(3 \cdot [\lim (x^2 + 2)]^4\)
Sum law & direct substitution: \(3 \cdot [(1 + 2)]^4 = 3 \cdot 3^4 = 3 \cdot 81 = 243\)

Example 4: Square Root

Find: \(\lim_{x \to 5} \sqrt{3x + 1}\)

Solution:

Root law: \(\lim \sqrt{3x + 1} = \sqrt{\lim (3x + 1)}\)
Direct substitution: \(\sqrt{3(5) + 1} = \sqrt{16} = 4\)

⚠️ Special Cases & When Laws Don't Apply

Limit laws FAIL when:

Denominator → 0: Quotient law doesn't apply if \(\lim g(x) = 0\)
Indeterminate forms: Getting \(\frac{0}{0}\), \(\frac{\infty}{\infty}\), \(0 \cdot \infty\), etc.
One or both limits DNE: Can't use laws if individual limits don't exist
Even root of negative: \(\sqrt{-4}\) is undefined in real numbers

What to Do When Direct Substitution Fails

If direct substitution gives you 0/0 or another indeterminate form, you need algebraic manipulation:

Factor and cancel (Unit 1.6)
Rationalize (multiply by conjugate)
Combine fractions
Use special limit theorems
L'Hôpital's Rule (later in calculus)

💡 Tips, Tricks & Mnemonics

✅ Essential Tips

ALWAYS try direct substitution first: It's the fastest method when it works
Check the denominator: Before using quotient law, verify it's not approaching zero
Work inside-out: For nested functions, evaluate innermost limits first
Factor when possible: If direct substitution fails, try factoring
Know your continuous functions: Polynomials, sin, cos, e^x, ln(x) are continuous on their domains

🎯 Mnemonic for Limit Laws

Memory Aid: "S-D-C-P-Q-P-R-C"

Sum
Difference
Constant Multiple
Product
Quotient
Power
Root
Composition

Or use: "Smart Cats Play Quietly, Producing Peaceful Relaxing Compositions"

🔍 Quick Diagnostic Flowchart

Finding a Limit? Follow This:

Is it continuous at that point? → YES: Use direct substitution (done!)
NO: Does direct substitution give 0/0?
- YES: Use algebraic manipulation (factor, rationalize)
- NO: Check if it gives ∞ or DNE → analyze further
Can you break it into pieces? → Use sum, difference, product, quotient laws
Is it a composition? → Check if outer function is continuous at inner limit

❌ Common Mistakes to Avoid

Mistake 1: Using quotient law when denominator → 0 (must factor first!)
Mistake 2: Forgetting to check continuity for composition law
Mistake 3: Canceling before verifying the factor doesn't make x = a undefined
Mistake 4: Applying laws when one or both individual limits don't exist
Mistake 5: Taking square root of a negative limit (undefined in reals)
Mistake 6: Not recognizing when direct substitution works—always try it first!

📝 How to Master These Properties

Effective Practice Strategy:

Memorize the laws: Use flashcards or the mnemonic
Start simple: Practice with polynomials first (always works)
Identify which laws apply: Before solving, name the properties you'll use
Work both directions: Given a limit, evaluate it; given a value, construct a matching limit
Check conditions: Always verify denominators aren't zero, roots are valid, etc.
Recognize when they don't apply: Practice identifying indeterminate forms
Time yourself: AP® exam limits should be quick with these laws

🌟 Special Limits to Know

Important Special Limits

These will be covered more in later units, but are worth noting:

\[ \lim_{x \to 0} \frac{\sin x}{x} = 1 \]

\[ \lim_{x \to 0} \frac{1 - \cos x}{x} = 0 \]

\[ \lim_{x \to \infty} \left(1 + \frac{1}{x}\right)^x = e \]

✏️ AP® Exam Tips

What the AP® Exam Expects:

Know which law to use: On FRQs, you may need to justify which property applies
Show your work: Write out the steps—don't just give the final answer
Check conditions: Explicitly verify denominators aren't zero
Use proper notation: Write \(\lim_{x \to a}\) clearly, not just "lim"
Direct substitution is your friend: Use it whenever possible—it's fastest
Multiple choice strategy: Eliminate answers that violate limit laws
Watch for traps: Problems that look simple but have hidden discontinuities

🔢 Special Case: Polynomials & Rational Functions

Polynomial Limit Theorem

For any polynomial \(p(x)\):

\[ \lim_{x \to a} p(x) = p(a) \]

Polynomials are continuous everywhere, so direct substitution always works!

Rational Function Limit Theorem

For rational function \(r(x) = \frac{p(x)}{q(x)}\):

\[ \lim_{x \to a} \frac{p(x)}{q(x)} = \frac{p(a)}{q(a)} \quad \text{(if } q(a) \neq 0\text{)} \]

If \(q(a) = 0\), you cannot use direct substitution—need algebraic manipulation!

⚡ Quick Reference Card

If You See...	Use This Law	Formula
\(f(x) + g(x)\)	Sum Law	\(\lim [f + g] = L + M\)
\(f(x) - g(x)\)	Difference Law	\(\lim [f - g] = L - M\)
\(c \cdot f(x)\)	Constant Multiple	\(\lim [c \cdot f] = c \cdot L\)
\(f(x) \cdot g(x)\)	Product Law	\(\lim [f \cdot g] = L \cdot M\)
\(\frac{f(x)}{g(x)}\)	Quotient Law	\(\lim \frac{f}{g} = \frac{L}{M}\) (\(M \neq 0\))
\([f(x)]^n\)	Power Law	\(\lim [f]^n = L^n\)
\(\sqrt[n]{f(x)}\)	Root Law	\(\lim \sqrt[n]{f} = \sqrt[n]{L}\)
\(f(g(x))\)	Composition	\(\lim f(g(x)) = f(L)\) (\(f\) continuous)

🔗 Connection to Other Topics

Unit 1.5 is foundational for:

Unit 1.6: Algebraic manipulation (when these properties aren't enough)
Unit 1.7-1.9: Continuity (uses limit properties to define continuous functions)
Unit 2: Derivatives (limit definition uses these properties)
Unit 3: Derivative rules (mirror these limit laws)
Unit 6: Integration properties (similar sum, difference, constant multiple rules)
All of calculus: These are the building blocks for everything!

Remember: The algebraic properties of limits are power tools that let you evaluate complex limits by breaking them into simple pieces. Master these nine laws (plus the composition rule), and you'll be able to tackle most limit problems with confidence. When direct substitution works, use it—it's fast! When it doesn't, Unit 1.6 has your back with algebraic manipulation techniques. 🎯✨