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Unit 1.14 – Connecting Infinite Limits and Vertical Asymptotes

AP® Calculus AB & BC | Formula Reference Sheet

Core Concept: When functions "blow up" and shoot off to infinity at certain x-values, we get vertical asymptotes. These dramatic behaviors are described mathematically using infinite limits. This unit connects the algebraic concept of limits approaching ±∞ with the graphical feature of vertical asymptote lines. Master this connection, and you'll understand exactly where and why functions become unbounded!

♾️ What Are Infinite Limits?

INFINITE LIMITS

An infinite limit occurs when a function \(f(x)\) grows without bound (increases or decreases infinitely) as \(x\) approaches a particular value \(a\).

Notation and Meaning:

Infinite Limit to Positive Infinity
\[ \lim_{x \to a} f(x) = +\infty \]

Meaning: As \(x\) approaches \(a\) from both sides, \(f(x)\) increases without bound—values become arbitrarily large and positive.

Infinite Limit to Negative Infinity
\[ \lim_{x \to a} f(x) = -\infty \]

Meaning: As \(x\) approaches \(a\) from both sides, \(f(x)\) decreases without bound—values become arbitrarily large and negative.

One-Sided Infinite Limits

From the right (right-hand limit):

\[ \lim_{x \to a^+} f(x) = \pm\infty \]

From the left (left-hand limit):

\[ \lim_{x \to a^-} f(x) = \pm\infty \]

Note: The one-sided limits can go to different infinities (one +∞, the other -∞) or the same infinity (both +∞ or both -∞).

📝 Important Distinction: When we write \(\lim_{x \to a} f(x) = \infty\), we're describing unbounded behavior, not saying the limit exists as a finite number. Technically, the limit does not exist (DNE) in the finite sense, but we use the infinity notation to precisely describe how it fails to exist!

📏 What Are Vertical Asymptotes?

VERTICAL ASYMPTOTE

A vertical asymptote is a vertical line \(x = a\) where the function grows without bound (approaches ±∞) as \(x\) approaches \(a\).

FORMAL DEFINITION: When Does x = a Have a Vertical Asymptote?

The line \(x = a\) is a vertical asymptote of \(f(x)\) if at least one of the following is true:

  1. \(\lim_{x \to a^+} f(x) = +\infty\)
  2. \(\lim_{x \to a^+} f(x) = -\infty\)
  3. \(\lim_{x \to a^-} f(x) = +\infty\)
  4. \(\lim_{x \to a^-} f(x) = -\infty\)

In other words: If the function "blows up" to ±∞ from at least one side, then \(x = a\) is a vertical asymptote!

💡 Key Insight: Vertical asymptotes represent infinite discontinuities—they're places where the function is not just undefined, but where it shoots off to infinity. The graph has a vertical "wall" that the function approaches but never crosses!

🔗 Connecting Infinite Limits and Vertical Asymptotes

THE FUNDAMENTAL CONNECTION

Infinite limits are the mathematical language we use to describe vertical asymptotes. They tell us how the function behaves near the asymptote.

  • Vertical Asymptote = Geometric feature (a vertical line)
  • Infinite Limit = Algebraic description (the limit notation)

To prove that \(x = a\) is a vertical asymptote, you must show that at least one one-sided limit approaches ±∞ using limit notation!

🔍 How to Find Vertical Asymptotes

STEP-BY-STEP PROCESS: Finding Vertical Asymptotes in Rational Functions

For \(f(x) = \frac{P(x)}{Q(x)}\) (rational function):

  1. STEP 1: Find where denominator = 0
    • Solve \(Q(x) = 0\) to find candidate x-values
    • These are potential vertical asymptotes
  2. STEP 2: Factor and check for cancellation
    • Factor both numerator \(P(x)\) and denominator \(Q(x)\) completely
    • Cancel any common factors
    • If a factor cancels: That point is a hole (removable discontinuity), NOT a vertical asymptote!
    • If a factor doesn't cancel: Continue to Step 3
  3. STEP 3: Check if numerator ≠ 0 at that point
    • Evaluate \(P(a)\) where \(a\) makes denominator = 0
    • If \(P(a) \neq 0\): Vertical asymptote at \(x = a\) ✓
    • If \(P(a) = 0\): You have \(\frac{0}{0}\)—might be a hole (check cancellation)
  4. STEP 4: Determine the sign (±∞) using one-sided limits
    • Evaluate \(\lim_{x \to a^-} f(x)\) and \(\lim_{x \to a^+} f(x)\)
    • Use sign analysis to determine if each side goes to +∞ or -∞
Quick Rule for Rational Functions

For \(f(x) = \frac{P(x)}{Q(x)}\), vertical asymptotes occur at values \(x = a\) where:

\[ Q(a) = 0 \quad \text{AND} \quad P(a) \neq 0 \]

(After canceling any common factors!)

➕➖ Sign Analysis: Determining +∞ or -∞

To determine whether the limit goes to +∞ or -∞:

  1. Factor the function and cancel common factors
  2. For each side of \(x = a\):
    • Pick a test value slightly to the left (for \(x \to a^-\))
    • Pick a test value slightly to the right (for \(x \to a^+\))
  3. Evaluate the sign of each factor at the test value
  4. Multiply the signs:
    • Positive result: Limit = +∞
    • Negative result: Limit = -∞

💡 Multiplicity Tip:

  • Odd multiplicity in denominator (e.g., \((x-a)^1\) or \((x-a)^3\)): One-sided limits have opposite signs (one +∞, other -∞)
  • Even multiplicity in denominator (e.g., \((x-a)^2\) or \((x-a)^4\)): One-sided limits have the same sign (both +∞ or both -∞)

📖 Comprehensive Worked Examples

Example 1: Basic Vertical Asymptote

Problem: Find all vertical asymptotes of \(f(x) = \frac{1}{x - 2}\)

Solution:

  1. Find where denominator = 0:
    • \(x - 2 = 0 \Rightarrow x = 2\)
  2. Check numerator at \(x = 2\):
    • Numerator = 1 ≠ 0 ✓
  3. Evaluate one-sided limits:
    • From the left: As \(x \to 2^-\), \(x - 2\) is negative and small
      \[ \lim_{x \to 2^-} \frac{1}{x-2} = -\infty \]
    • From the right: As \(x \to 2^+\), \(x - 2\) is positive and small
      \[ \lim_{x \to 2^+} \frac{1}{x-2} = +\infty \]

Conclusion: Vertical asymptote at \(x = 2\) ✓

Behavior: Function goes to -∞ from left, +∞ from right

Example 2: Removable vs. Non-Removable

Problem: Find vertical asymptotes of \(g(x) = \frac{x^2 - 9}{x^2 - 5x + 6}\)

Solution:

  1. Factor numerator and denominator:
    • \(x^2 - 9 = (x - 3)(x + 3)\)
    • \(x^2 - 5x + 6 = (x - 2)(x - 3)\)
  2. Rewrite:
    \[ g(x) = \frac{(x-3)(x+3)}{(x-2)(x-3)} \]
  3. Cancel common factor \((x-3)\):
    \[ g(x) = \frac{x+3}{x-2} \quad (x \neq 3) \]
  4. At \(x = 3\): Factor canceled → Hole (removable), NOT a vertical asymptote
  5. At \(x = 2\): Denominator still = 0, numerator = 5 ≠ 0
    • \(\lim_{x \to 2^-} \frac{x+3}{x-2} = -\infty\)
    • \(\lim_{x \to 2^+} \frac{x+3}{x-2} = +\infty\)

Conclusion:

  • Vertical asymptote at \(x = 2\) ✓
  • Removable discontinuity (hole) at \(x = 3\)

Example 3: Even Multiplicity

Problem: Find vertical asymptotes of \(h(x) = \frac{3}{(x + 1)^2}\)

Solution:

  1. Denominator = 0: \((x + 1)^2 = 0 \Rightarrow x = -1\)
  2. Numerator at \(x = -1\): 3 ≠ 0 ✓
  3. Note even multiplicity: \((x + 1)^2\) has exponent 2 (even)
  4. Both sides have same sign:
    • From left: \((x+1)^2\) is positive (squared) and small → \(\frac{3}{\text{small positive}} = +\infty\)
    • From right: \((x+1)^2\) is positive (squared) and small → \(\frac{3}{\text{small positive}} = +\infty\)
  5. Limits:
    \[ \lim_{x \to -1^-} h(x) = +\infty \quad \text{and} \quad \lim_{x \to -1^+} h(x) = +\infty \]

Conclusion: Vertical asymptote at \(x = -1\), both sides go to +∞

Example 4: Natural Logarithm

Problem: Show that \(f(x) = \ln(x)\) has a vertical asymptote at \(x = 0\)

Solution:

  1. Domain of \(\ln(x)\): \(x > 0\) only
  2. Evaluate limit from the right:
    • As \(x \to 0^+\), \(\ln(x) \to -\infty\)
    • (\(e^n \to 0\) requires very negative \(n\))
  3. Left-hand limit doesn't exist: Function undefined for \(x \leq 0\)
  4. Write limit:
    \[ \lim_{x \to 0^+} \ln(x) = -\infty \]

Conclusion: Vertical asymptote at \(x = 0\) (y-axis) ✓

Example 5: Multiple Asymptotes

Problem: Find all vertical asymptotes of \(p(x) = \frac{2x}{x^2 - 4}\)

Solution:

  1. Factor denominator: \(x^2 - 4 = (x - 2)(x + 2)\)
  2. Denominator = 0: \(x = 2\) or \(x = -2\)
  3. Check numerator: \(2x\) at \(x = 2\) is 4 ≠ 0; at \(x = -2\) is -4 ≠ 0 ✓
  4. At \(x = 2\):
    • Test \(x = 1.9\): \(\frac{3.8}{(-0.1)(3.9)} < 0\) → -∞
    • Test \(x = 2.1\): \(\frac{4.2}{(0.1)(4.1)} > 0\) → +∞
  5. At \(x = -2\):
    • Test \(x = -2.1\): \(\frac{-4.2}{(-4.1)(-0.1)} < 0\) → -∞
    • Test \(x = -1.9\): \(\frac{-3.8}{(-3.9)(0.1)} > 0\) → +∞

Conclusion: Two vertical asymptotes at \(x = 2\) and \(x = -2\)

📊 Removable vs. Non-Removable Discontinuities

Feature Removable (Hole) Vertical Asymptote
Cause Common factor cancels Denominator = 0, no cancellation
Limit behavior Limit exists (finite) Limit = ±∞ (infinite)
Graphical Open circle (hole) Vertical dashed line
Can cross? N/A (single point) NO—function never crosses
Fixable? ✓ YES (redefine point) ✗ NO (unbounded behavior)
Example \(\frac{x^2-4}{x-2}\) at \(x=2\) \(\frac{1}{x-2}\) at \(x=2\)
Limit notation \(\lim_{x \to a} f(x) = L\) (finite) \(\lim_{x \to a^\pm} f(x) = \pm\infty\)

📚 Common Functions with Vertical Asymptotes

Function Vertical Asymptote(s) Behavior
\(f(x) = \frac{1}{x}\) \(x = 0\) Left: -∞, Right: +∞
\(f(x) = \frac{1}{x^2}\) \(x = 0\) Both sides: +∞
\(f(x) = \ln(x)\) \(x = 0\) Right only: -∞
\(f(x) = \tan(x)\) \(x = \frac{\pi}{2} + n\pi\) Alternating ±∞
\(f(x) = \cot(x)\) \(x = n\pi\) Alternating ±∞
\(f(x) = \sec(x)\) \(x = \frac{\pi}{2} + n\pi\) Alternating ±∞
\(f(x) = \csc(x)\) \(x = n\pi\) Alternating ±∞

💡 Tips, Tricks & Strategies

✅ Essential Tips

  • Always factor first: Cancel common factors before identifying vertical asymptotes
  • Check both sides: Evaluate both one-sided limits (left and right)
  • Use test values: Pick numbers slightly left and right of the asymptote for sign analysis
  • Remember multiplicity: Even powers → same sign both sides; Odd powers → opposite signs
  • Don't confuse with holes: If factors cancel, it's removable (hole), not an asymptote
  • Graph can't cross VA: Vertical asymptotes are never crossed (unlike horizontal asymptotes)
  • State direction: Specify whether the limit goes to +∞ or -∞ from each side

🎯 The "FACD" Method

Quick memory aid for finding vertical asymptotes:

  • Factor — Factor numerator and denominator completely
  • Anything cancel? — Look for and cancel common factors
  • Check denominator — Find where remaining denominator = 0
  • Determine signs — Use test values to find ±∞ for each side

🔥 Quick Recognition Patterns

You likely have a vertical asymptote if you see:

  • \(\frac{\text{nonzero}}{0}\) after substitution (and no cancellation)
  • Denominator = 0 but numerator ≠ 0 at that point
  • After canceling, denominator still has zeros
  • Graph shoots to ±∞ near that x-value

You have a hole (NOT asymptote) if:

  • \(\frac{0}{0}\) after substitution
  • Common factor cancels from numerator and denominator
  • After canceling, denominator ≠ 0 at that point
  • Graph has open circle at that point

❌ Common Mistakes to Avoid

  • Mistake 1: Calling every zero of the denominator a vertical asymptote (forget to check cancellation!)
  • Mistake 2: Not checking one-sided limits separately—they can be different!
  • Mistake 3: Forgetting to factor before identifying asymptotes
  • Mistake 4: Confusing vertical asymptotes (infinite) with holes (removable)
  • Mistake 5: Not specifying +∞ or -∞—the sign matters!
  • Mistake 6: Saying "limit = ∞" without clarifying from which side
  • Mistake 7: Thinking the function can cross a vertical asymptote (it can't!)
  • Mistake 8: Ignoring multiplicity effects on sign analysis

📝 Practice Problems

Find all vertical asymptotes for each function:

  1. \(f(x) = \frac{x + 5}{x - 3}\)
  2. \(g(x) = \frac{x^2 - 16}{x - 4}\)
  3. \(h(x) = \frac{2x + 1}{x^2 - 1}\)
  4. \(p(x) = \frac{x}{(x + 2)^2}\)

Answers:

  1. \(x = 3\) — Vertical asymptote (left: -∞, right: +∞)
  2. No vertical asymptote — Hole at \(x = 4\) (factors cancel)
  3. \(x = 1\) and \(x = -1\) — Two vertical asymptotes
  4. \(x = -2\) — Even multiplicity (both sides → same ∞)

✏️ AP® Exam Success Tips

What the AP® Exam Expects:

  • Use proper limit notation: Write \(\lim_{x \to a^\pm} f(x) = \pm\infty\) explicitly
  • Show factorization: Factor to check for cancellation—show all steps
  • State one-sided behavior: Specify whether each side goes to +∞ or -∞
  • Justify with limits: Prove asymptotes exist using infinite limit statements
  • Distinguish from holes: Explain why it's not a removable discontinuity
  • Use proper terminology: Say "vertical asymptote," not just "undefined"
  • Graph sketches: Draw vertical dashed lines for asymptotes on coordinate planes

Common FRQ Formats:

  1. "Find all vertical asymptotes and justify using limits"
  2. "Using limit notation, show that x = a is a vertical asymptote"
  3. "Determine the behavior of f(x) as x approaches a from each side"
  4. "Sketch the graph showing all vertical asymptotes"
  5. "Explain why the discontinuity at x = a is non-removable"

⚡ Quick Reference Card

Step Action What to Look For
1. Factor Factor numerator & denominator Common factors to cancel
2. Cancel Remove common factors Holes (removable) vs. asymptotes
3. Find zeros Set remaining denominator = 0 Candidate asymptote locations
4. Check numerator Evaluate numerator at those points Must be nonzero for VA
5. Sign analysis Test values left and right Determine +∞ or -∞

🔗 Why This Unit Matters

Unit 1.14 connects to:

  • Unit 1.10: Types of discontinuities (vertical asymptotes are infinite discontinuities)
  • Unit 1.15: Limits at infinity and horizontal asymptotes (complementary concept)
  • Unit 3: Analyzing function behavior and sketching accurate graphs
  • Unit 4: Curve sketching requires identifying all asymptotes
  • Unit 6: Improper integrals with infinite discontinuities
  • Real-world: Modeling situations with unbounded behavior (physics, economics)

Remember: Infinite limits describe the mathematical behavior (how the function "blows up"), while vertical asymptotes describe the geometric feature (the vertical line the graph approaches). They're two sides of the same coin! To find vertical asymptotes: Factor, Cancel, Check denominator zeros, Determine signs (FACD). Always use proper limit notation with ±∞ to justify asymptotes on the AP® exam. Master this connection, and you'll understand exactly where and why functions become unbounded! 🚀✨