Solved Examples: Finding Vertical Asymptotes of Rational Functions

📚 Quick Study Guide

• 🔍 A rational function is a function that can be defined as a fraction $\frac{P(x)}{Q(x)}$, where both $P(x)$ and $Q(x)$ are polynomials.
• 💡 A vertical asymptote occurs at $x = a$ if the limit of the function as $x$ approaches $a$ is infinite (either positive or negative). Formally, $\lim_{x \to a} f(x) = \pm \infty$.
• 📝 To find vertical asymptotes:
  1. Set the denominator $Q(x)$ equal to zero and solve for $x$.
  2. Check if the solutions also make the numerator $P(x)$ equal to zero. If they do, there might be a hole instead of a vertical asymptote. Simplify the rational function if possible.
  3. If $Q(a) = 0$ and $P(a) \neq 0$, then $x = a$ is a vertical asymptote.
• 🧮 Remember to simplify the rational function first by canceling out any common factors in the numerator and denominator. This helps in identifying holes versus vertical asymptotes.
• 📈 Vertical asymptotes are represented as vertical dashed lines on the graph of the function, showing where the function approaches infinity.

Practice Quiz

1. What is the first step in finding the vertical asymptote(s) of a rational function?
1. Set the entire rational function equal to zero.
2. Set the numerator equal to zero.
3. Set the denominator equal to zero.
4. Find the derivative of the function.


2. For the rational function $f(x) = \frac{x+2}{x-3}$, what is the vertical asymptote?
1. $x = -2$
2. $x = 3$
3. $x = 2$
4. $x = -3$


3. Consider the function $f(x) = \frac{x-1}{(x-1)(x+2)}$. What is the vertical asymptote?
1. $x = 1$
2. $x = -2$
3. Both $x = 1$ and $x = -2$
4. There is no vertical asymptote.


4. Which of the following functions has a vertical asymptote at $x = 0$?
1. $f(x) = \frac{x}{x+1}$
2. $f(x) = \frac{x+1}{x}$
3. $f(x) = \frac{1}{x^2+1}$
4. $f(x) = \frac{x^2+1}{x+1}$


5. What happens if a value makes both the numerator and denominator of a rational function equal to zero?
1. It is always a vertical asymptote.
2. It indicates a hole in the graph.
3. It indicates a horizontal asymptote.
4. It means the function is undefined everywhere.


6. For the function $f(x) = \frac{x^2 - 4}{x - 2}$, what exists at $x = 2$?
1. A vertical asymptote
2. A hole
3. A horizontal asymptote
4. An x-intercept


7. Which of the following statements is true about vertical asymptotes?
1. A function can cross a vertical asymptote.
2. A function can never cross a vertical asymptote.
3. Vertical asymptotes are always at $x = 0$.
4. Vertical asymptotes indicate where the function equals zero.