How to find horizontal asymptotes using limits (examples)

📚 Quick Study Guide

• 📈 Definition: A horizontal asymptote is a horizontal line that the graph of a function approaches as $x$ tends to positive or negative infinity.
• 🧮 Finding Horizontal Asymptotes: To find horizontal asymptotes, evaluate the following limits: $\lim_{x \to \infty} f(x)$ and $\lim_{x \to -\infty} f(x)$.
• ⚖️ Cases:
  • If $\lim_{x \to \infty} f(x) = L$ or $\lim_{x \to -\infty} f(x) = L$, then $y = L$ is a horizontal asymptote.
  • If the limit does not exist (DNE) or is infinite, then there is no horizontal asymptote in that direction.
• 💡 Rational Functions: For rational functions ($\frac{P(x)}{Q(x)}$):
  • If degree of $P(x)$ < degree of $Q(x)$, then $y = 0$ is a horizontal asymptote.
  • If degree of $P(x)$ = degree of $Q(x)$, then $y = \frac{\text{leading coefficient of } P(x)}{\text{leading coefficient of } Q(x)}$ is a horizontal asymptote.
  • If degree of $P(x)$ > degree of $Q(x)$, then there is no horizontal asymptote.

Practice Quiz

1. 1. What is the horizontal asymptote of $f(x) = \frac{2x}{x+1}$?

   1. A. $y = 0$
   2. B. $y = 1$
   3. C. $y = 2$
   4. D. $y = -1$

2. 2. What is the horizontal asymptote of $f(x) = \frac{3x^2 + 1}{x^3}$?

   1. A. $y = 0$
   2. B. $y = 1$
   3. C. $y = 3$
   4. D. No horizontal asymptote

3. 3. What is the horizontal asymptote of $f(x) = \frac{x^2 + 2}{2x^2 - 1}$?

   1. A. $y = 0$
   2. B. $y = 1$
   3. C. $y = \frac{1}{2}$
   4. D. No horizontal asymptote

4. 4. What is the horizontal asymptote of $f(x) = \frac{e^x}{x}$ as $x$ approaches $-\infty$?

   1. A. $y = 0$
   2. B. $y = 1$
   3. C. $y = \infty$
   4. D. Does Not Exist

5. 5. What is the horizontal asymptote of $f(x) = \frac{\sqrt{x^2 + 1}}{x}$ as $x$ approaches $\infty$?

   1. A. $y = 0$
   2. B. $y = 1$
   3. C. $y = -1$
   4. D. No horizontal asymptote

6. 6. What is the horizontal asymptote of $f(x) = \arctan(x)$ as $x$ approaches $\infty$?

   1. A. $y = 0$
   2. B. $y = \frac{\pi}{2}$
   3. C. $y = \pi$
   4. D. No horizontal asymptote

7. 7. What is the horizontal asymptote of $f(x) = \frac{5x^3 - 2x + 1}{x^4 + x^2 - 3}$?

   1. A. $y = 5$
   2. B. $y = 0$
   3. C. $y = \frac{5}{4}$
   4. D. No horizontal asymptote