Detailed examples of horizontal asymptote rules and calculations

📚 Quick Study Guide

• 📈 Horizontal asymptotes describe the behavior of a function as $x$ approaches positive or negative infinity.
• 📏 Rule 1 (Degree of Numerator < Degree of Denominator): If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is $y = 0$.
• ➗ Rule 2 (Degree of Numerator = Degree of Denominator): If the degree of the numerator is equal to the degree of the denominator, the horizontal asymptote is $y = \frac{\text{leading coefficient of numerator}}{\text{leading coefficient of denominator}}$.
• 🚀 Rule 3 (Degree of Numerator > Degree of Denominator): If the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote (there may be a slant/oblique asymptote).
• 💡 To find a horizontal asymptote, analyze the function $f(x)$ as $x \rightarrow \infty$ and $x \rightarrow -\infty$.

Practice Quiz

1. Question 1: What is the horizontal asymptote of the function $f(x) = \frac{3x}{x^2 + 1}$?
   1. y = 3
   2. y = 0
   3. y = 1
   4. No horizontal asymptote
2. Question 2: What is the horizontal asymptote of the function $f(x) = \frac{4x^2 + 2x - 1}{2x^2 - 5}$?
   1. y = 0
   2. y = 2
   3. y = 4
   4. No horizontal asymptote
3. Question 3: What is the horizontal asymptote of the function $f(x) = \frac{x^3 + 1}{x^2 - 4}$?
   1. y = 0
   2. y = 1
   3. y = 3
   4. No horizontal asymptote
4. Question 4: What is the horizontal asymptote of the function $f(x) = \frac{5x - 3}{x + 2}$?
   1. y = 5
   2. y = 0
   3. y = -2
   4. No horizontal asymptote
5. Question 5: What is the horizontal asymptote of the function $f(x) = \frac{7}{x^3 + 2x}$?
   1. y = 7
   2. y = 0
   3. y = 1
   4. No horizontal asymptote
6. Question 6: Determine the horizontal asymptote of the function $f(x) = \frac{-2x^2 + 5}{3x^2 - x}$?
   1. y = 0
   2. y = -2/3
   3. y = 5/(-1)
   4. No horizontal asymptote
7. Question 7: Does the function $f(x) = \frac{x^4 - 1}{x^4 + 1}$ have a horizontal asymptote? If so, what is it?
   1. No horizontal asymptote
   2. y = 0
   3. y = 1
   4. y = -1