Unit 1.15 – Connecting Limits at Infinity and Horizontal Asymptotes

AP® Calculus AB & BC | Formula Reference Sheet

Core Concept: While vertical asymptotes describe what happens when functions "blow up" at specific x-values, horizontal asymptotes describe what happens at the "ends" of a graph—what y-value the function approaches as x goes to ±∞. Limits at infinity are the mathematical language we use to precisely describe this end behavior. Master this connection, and you'll understand exactly where functions "settle down" at their extremes!

♾️ What Are Limits at Infinity?

LIMITS AT INFINITY

A limit at infinity describes the value that a function \(f(x)\) approaches as \(x\) grows without bound in the positive or negative direction.

Limit as x Approaches Positive Infinity

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

Meaning: As \(x\) becomes arbitrarily large (increases without bound), the function values \(f(x)\) get arbitrarily close to the number \(L\).

In other words: The further right you go on the graph, the closer the y-values get to \(L\).

Limit as x Approaches Negative Infinity

\[ \lim_{x \to -\infty} f(x) = M \]

Meaning: As \(x\) becomes arbitrarily large in the negative direction (decreases without bound), the function values \(f(x)\) get arbitrarily close to the number \(M\).

In other words: The further left you go on the graph, the closer the y-values get to \(M\).

📝 Key Insight: Limits at infinity describe end behavior—what happens to the function at the "edges" of the graph. Note that \(L\) and \(M\) can be different, meaning the function can approach different values on the left and right!

📏 What Are Horizontal Asymptotes?

HORIZONTAL ASYMPTOTE

A horizontal asymptote is a horizontal line \(y = L\) that the graph of \(f(x)\) approaches as \(x\) goes to ±∞.

Formal Definition: The line \(y = L\) is a horizontal asymptote of \(f(x)\) if:

\[ \lim_{x \to \infty} f(x) = L \quad \text{or} \quad \lim_{x \to -\infty} f(x) = L \]

(Or both!)

💡 Important Distinction: Unlike vertical asymptotes (which the graph can NEVER cross), a function CAN cross a horizontal asymptote! The horizontal asymptote only describes what happens at the "ends" (as \(x \to \pm\infty\)). The function can cross \(y = L\) finitely or even infinitely many times in the middle!

🔗 Connecting Limits at Infinity and Horizontal Asymptotes

THE FUNDAMENTAL CONNECTION

Limits at infinity are the mathematical tool we use to find and justify horizontal asymptotes. They tell us precisely what value the function approaches at the ends.

Horizontal Asymptote = Geometric feature (a horizontal line \(y = L\))
Limit at Infinity = Algebraic description (the limit notation)

To prove that \(y = L\) is a horizontal asymptote, you must show that \(\lim_{x \to \infty} f(x) = L\) or \(\lim_{x \to -\infty} f(x) = L\) using limit notation!

How Many Horizontal Asymptotes Can a Function Have?

At most TWO: One as \(x \to \infty\) and one as \(x \to -\infty\)
The two horizontal asymptotes can be the same line (most common) or different lines
A function might have no horizontal asymptotes if limits at infinity don't exist or are infinite

📐 The Three Rules for Finding Horizontal Asymptotes

For Rational Functions: \(f(x) = \frac{P(x)}{Q(x)}\)

Where \(P(x)\) and \(Q(x)\) are polynomials.

🥇 RULE 1: Degree of Numerator < Degree of Denominator

Condition: deg(P) < deg(Q)

\[ \lim_{x \to \pm\infty} \frac{P(x)}{Q(x)} = 0 \]

Horizontal Asymptote: \(y = 0\) (the x-axis)

Why? The denominator grows faster than the numerator, so the fraction approaches zero.

🥈 RULE 2: Degree of Numerator = Degree of Denominator

Condition: deg(P) = deg(Q)

\[ \lim_{x \to \pm\infty} \frac{P(x)}{Q(x)} = \frac{\text{leading coefficient of } P}{\text{leading coefficient of } Q} \]

Horizontal Asymptote: \(y = \frac{a}{b}\) (ratio of leading coefficients)

Why? The highest-degree terms dominate, so the ratio simplifies to the ratio of their coefficients.

🥉 RULE 3: Degree of Numerator > Degree of Denominator

Condition: deg(P) > deg(Q)

\[ \lim_{x \to \pm\infty} \frac{P(x)}{Q(x)} = \pm\infty \]

Horizontal Asymptote: NONE (no horizontal asymptote)

Why? The numerator grows faster, so the function grows without bound.

Special Note: If deg(P) = deg(Q) + 1, you get a slant (oblique) asymptote instead!

📖 Comprehensive Worked Examples

Example 1: Rule 1 (Numerator Degree < Denominator Degree)

Problem: Find \(\lim_{x \to \infty} \frac{5x + 3}{2x^2 - x + 1}\) and identify the horizontal asymptote

Solution:

Compare degrees:
- Numerator: degree 1
- Denominator: degree 2
- 1 < 2 → Apply Rule 1
Alternative method (divide by highest power):
- Divide numerator and denominator by \(x^2\):
\[ \frac{5x + 3}{2x^2 - x + 1} = \frac{\frac{5}{x} + \frac{3}{x^2}}{2 - \frac{1}{x} + \frac{1}{x^2}} \]
Take the limit:
- As \(x \to \infty\): \(\frac{5}{x} \to 0\), \(\frac{3}{x^2} \to 0\), \(\frac{1}{x} \to 0\), \(\frac{1}{x^2} \to 0\)
\[ \lim_{x \to \infty} \frac{0 + 0}{2 - 0 + 0} = \frac{0}{2} = 0 \]

Answer: \(\lim_{x \to \infty} f(x) = 0\)

Horizontal Asymptote: \(y = 0\) ✓

Example 2: Rule 2 (Equal Degrees)

Problem: Find \(\lim_{x \to -\infty} \frac{3x^2 - 4x + 7}{6x^2 + 5}\) and identify the horizontal asymptote

Solution:

Compare degrees:
- Numerator: degree 2 (leading term \(3x^2\))
- Denominator: degree 2 (leading term \(6x^2\))
- Degrees equal → Apply Rule 2
Take ratio of leading coefficients:
\[ \frac{3}{6} = \frac{1}{2} \]
Alternative verification (divide by \(x^2\)):
\[ \frac{3x^2 - 4x + 7}{6x^2 + 5} = \frac{3 - \frac{4}{x} + \frac{7}{x^2}}{6 + \frac{5}{x^2}} \to \frac{3}{6} = \frac{1}{2} \]

Answer: \(\lim_{x \to -\infty} f(x) = \frac{1}{2}\)

Horizontal Asymptote: \(y = \frac{1}{2}\) ✓

Example 3: Rule 3 (Numerator Degree > Denominator Degree)

Problem: Find \(\lim_{x \to \infty} \frac{2x^3 + x - 1}{x^2 + 5}\) and determine if there's a horizontal asymptote

Solution:

Compare degrees:
- Numerator: degree 3
- Denominator: degree 2
- 3 > 2 → Apply Rule 3
Analyze behavior:
- Divide by \(x^2\):
\[ \frac{2x^3 + x - 1}{x^2 + 5} = \frac{2x + \frac{1}{x} - \frac{1}{x^2}}{1 + \frac{5}{x^2}} \]
As \(x \to \infty\):
- Numerator: \(2x \to \infty\)
- Denominator: \(1 + 0 = 1\)
- Quotient: \(\frac{\infty}{1} = \infty\)

Answer: \(\lim_{x \to \infty} f(x) = \infty\)

Horizontal Asymptote: NONE (function grows without bound)

Example 4: Polynomial Function

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

Solution:

Identify dominant term: \(2x^3\) (highest degree)
As \(x \to \infty\): The \(2x^3\) term dominates
- \(5x\) and \(-1\) become negligible compared to \(2x^3\)
Conclusion: \(2x^3 \to +\infty\) as \(x \to \infty\)

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

Horizontal Asymptote: NONE

Example 5: Exponential in Denominator

Problem: Find \(\lim_{x \to \infty} \frac{3x - 1}{e^x}\)

Solution:

Compare growth rates:
- Numerator: polynomial (linear)
- Denominator: exponential \(e^x\)
- Key fact: Exponentials grow MUCH faster than polynomials
As \(x \to \infty\):
- Numerator \(3x\) grows linearly
- Denominator \(e^x\) grows exponentially → much faster
- Fraction: \(\frac{\text{small}}{\text{huge}} \to 0\)

Answer: \(\lim_{x \to \infty} \frac{3x - 1}{e^x} = 0\)

Horizontal Asymptote: \(y = 0\) ✓

Example 6: Different Asymptotes Left and Right

Problem: Find the horizontal asymptotes of \(f(x) = \frac{x}{\sqrt{x^2 + 1}}\)

Solution:

As \(x \to \infty\):
- Factor \(x^2\) from the square root:
\[ \sqrt{x^2 + 1} = \sqrt{x^2(1 + \frac{1}{x^2})} = |x|\sqrt{1 + \frac{1}{x^2}} \]
- For \(x > 0\): \(|x| = x\)
\[ \frac{x}{x\sqrt{1 + \frac{1}{x^2}}} = \frac{1}{\sqrt{1 + \frac{1}{x^2}}} \to \frac{1}{1} = 1 \]
As \(x \to -\infty\):
- For \(x < 0\): \(|x| = -x\)
\[ \frac{x}{-x\sqrt{1 + \frac{1}{x^2}}} = \frac{-1}{\sqrt{1 + \frac{1}{x^2}}} \to -1 \]

Horizontal Asymptotes:

\(y = 1\) as \(x \to \infty\)
\(y = -1\) as \(x \to -\infty\)

🌱 Growth Rates of Functions

Understanding which functions grow faster is crucial for limits at infinity!

Growth Rate Hierarchy (from slowest to fastest):

\[ \text{Logarithmic} < \text{Root} < \text{Polynomial} < \text{Exponential} \]

In symbols:

\[ \ln(x) < \sqrt{x} < x^n < e^x < x! \]

Key Growth Rate Facts:

Exponentials dominate polynomials: \(\lim_{x \to \infty} \frac{x^n}{e^x} = 0\) for any \(n\)
Polynomials dominate logarithms: \(\lim_{x \to \infty} \frac{\ln(x)}{x^n} = 0\) for any \(n > 0\)
Among polynomials, higher degree wins: \(\lim_{x \to \infty} \frac{x^m}{x^n} = \begin{cases} 0 & m < n \\ 1 & m = n \\ \infty & m > n \end{cases}\)

⚠️ Special Cases and Exceptions

1. Oscillating Functions

Functions like \(\sin(x)\) and \(\cos(x)\) oscillate forever and don't approach a limit.

\[ \lim_{x \to \infty} \sin(x) = \text{DNE (does not exist)} \]

No horizontal asymptote!

2. Bounded Oscillating Functions

If an oscillating function is divided by something that grows, it can approach zero.

\[ \lim_{x \to \infty} \frac{\sin(x)}{x} = 0 \]

Horizontal asymptote: \(y = 0\)

Why? Squeeze Theorem: \(-1 \leq \sin(x) \leq 1\), so \(\frac{-1}{x} \leq \frac{\sin(x)}{x} \leq \frac{1}{x}\), both ends → 0

📊 Quick Reference: The Three Rules

Rule	Condition	Limit	Horizontal Asymptote
Rule 1	deg(num) < deg(den)	\(\lim_{x \to \pm\infty} f(x) = 0\)	\(y = 0\)
Rule 2	deg(num) = deg(den)	\(\lim_{x \to \pm\infty} f(x) = \frac{a}{b}\)	\(y = \frac{a}{b}\)
Rule 3	deg(num) > deg(den)	\(\lim_{x \to \pm\infty} f(x) = \pm\infty\)	NONE

↕️ Vertical vs. Horizontal Asymptotes

Feature	Vertical Asymptote	Horizontal Asymptote
Line equation	\(x = a\) (vertical line)	\(y = L\) (horizontal line)
Limit type	Infinite limit: \(\lim_{x \to a} f(x) = \pm\infty\)	Limit at infinity: \(\lim_{x \to \pm\infty} f(x) = L\)
Describes	Behavior near \(x = a\) (local)	End behavior as \(x \to \pm\infty\) (global)
Can cross?	✗ NO (never)	✓ YES (can cross multiple times)
Maximum number	Unlimited (many possible)	At most 2 (one each direction)
Common cause	Denominator = 0	Degree comparison in rational functions

💡 Tips, Tricks & Strategies

✅ Essential Tips

Always compare degrees first: For rational functions, this instantly tells you which rule to use
Divide by highest power: When in doubt, divide numerator and denominator by the highest power of \(x\)
Remember growth rates: Exponential > Polynomial > Logarithmic
Check both directions: Evaluate \(\lim_{x \to \infty}\) and \(\lim_{x \to -\infty}\) separately
Functions can cross horizontal asymptotes: Unlike vertical asymptotes!
Use L'Hôpital's Rule if needed: For \(\frac{\infty}{\infty}\) or \(\frac{0}{0}\) indeterminate forms

🎯 The "Divide by Highest Power" Method

Universal technique for rational functions:

Identify the highest power of \(x\) in the denominator
Divide every term (numerator and denominator) by that power
Apply \(\lim_{x \to \infty} \frac{1}{x^n} = 0\) to simplify
The result is your limit (and horizontal asymptote if finite)!

🔥 Quick Recognition Patterns

Instant identification:

See \(e^x\) in denominator? → Limit = 0 (exponential dominates)
Same degree, coefficients 3 and 5? → HA is \(y = \frac{3}{5}\)
Numerator degree > denominator? → No horizontal asymptote (grows to ±∞)
See \(\frac{\sin(x)}{x}\)? → Limit = 0 by Squeeze Theorem
Polynomial alone (no fraction)? → Limit = ±∞ (no HA unless constant)

❌ Common Mistakes to Avoid

Mistake 1: Thinking horizontal asymptotes can't be crossed (they can!)
Mistake 2: Only checking \(x \to \infty\) and forgetting \(x \to -\infty\)
Mistake 3: Using the constant term instead of leading coefficient in Rule 2
Mistake 4: Forgetting to factor \(|x|\) correctly with square roots (sign matters!)
Mistake 5: Saying "limit equals infinity" when it doesn't exist (be precise: "approaches" or "is infinite")
Mistake 6: Confusing vertical asymptotes (local, can't cross) with horizontal (global, can cross)
Mistake 7: Not recognizing exponential dominance over polynomials
Mistake 8: Dividing by the wrong power of \(x\) (use highest power in denominator)

📝 Practice Problems

Find the horizontal asymptotes (if any) for each function:

\(f(x) = \frac{4x - 7}{2x + 3}\)
\(g(x) = \frac{x^2 + 1}{3x^3 - 2}\)
\(h(x) = \frac{5x^3 + 2x}{x^2 - 4}\)
\(p(x) = \frac{2^x}{x^2}\)

Answers:

\(y = 2\) — Rule 2 (equal degrees): \(\frac{4}{2} = 2\)
\(y = 0\) — Rule 1 (deg 2 < deg 3)
No HA — Rule 3 (deg 3 > deg 2): limit = ±∞
No HA — Exponential in numerator dominates; limit = ∞

✏️ AP® Exam Success Tips

What the AP® Exam Expects:

Use proper limit notation: Write \(\lim_{x \to \infty} f(x) = L\) explicitly
Justify using the three rules: State which degree condition applies
Show your work: Demonstrate dividing by highest power or comparing degrees
State the asymptote equation: Write "The horizontal asymptote is \(y = L\)"
Check both directions: Verify \(x \to \infty\) and \(x \to -\infty\)
Distinguish from vertical asymptotes: Make clear which type you're finding
Use correct terminology: "End behavior," "approaches," "limit at infinity"

Common FRQ Formats:

"Find all horizontal asymptotes of f and justify using limits"
"Describe the end behavior of f as x approaches infinity"
"Using limit notation, show that y = L is a horizontal asymptote"
"Determine \(\lim_{x \to \infty} f(x)\) and \(\lim_{x \to -\infty} f(x)\)"
"Sketch the graph showing all horizontal and vertical asymptotes"

⚡ Quick Reference Card

Quick Check	What to Do	Result
Rational function	Compare degrees	Use Rule 1, 2, or 3
Polynomial alone	Check leading term	Limit = ±∞ (no HA)
Exponential present	Compare growth rates	Exp dominates polynomial
Oscillating (sin/cos)	Check if bounded	Often DNE or use Squeeze
With square root	Factor \|x\| from radical	Check sign for ±∞

🔗 Why This Unit Matters

Unit 1.15 connects to:

Unit 1.14: Complements vertical asymptotes (x → a vs. x → ±∞)
Unit 3-4: Curve sketching requires identifying all asymptotes
Unit 6: Improper integrals and convergence depend on end behavior
Unit 8 (BC): Series convergence tests use limits at infinity
Unit 10 (BC): Parametric and polar functions have end behavior too
Real-world: Long-term trends, carrying capacity, steady-state behavior

Remember: Limits at infinity describe end behavior (what happens as x → ±∞), while horizontal asymptotes are the horizontal lines y = L that result from these limits. To find horizontal asymptotes of rational functions: compare degrees (Rule 1: deg(num) < deg(den) → y = 0; Rule 2: equal degrees → y = ratio; Rule 3: deg(num) > deg(den) → no HA). Always use proper limit notation to justify asymptotes on the AP® exam. Master the three rules, understand growth rates, and remember: functions CAN cross their horizontal asymptotes! 🎯✨