AP Calculus AB 1.14 Finding the Vertical Asymptotes of a Rational Function

In AP Calculus AB, topic 1.14 focuses on finding the vertical asymptotes of a rational function. This topic is essential for understanding the behavior of functions as they approach certain critical values, where the function becomes undefined. Here’s a detailed breakdown:

### Key Concepts:

1. **Vertical Asymptotes:**
- A vertical asymptote occurs at \( x = a \) for a function \( f(x) \) if the function approaches infinity (or negative infinity) as \( x \) approaches \( a \). This is typically indicated by:
\[
\lim_{x \to a} f(x) = \pm \infty.
\]

2. **Rational Functions:**
- A rational function is expressed as the ratio of two polynomials:
\[
f(x) = \frac{P(x)}{Q(x)},
\]
where \( P(x) \) and \( Q(x) \) are polynomials.

3. **Identifying Vertical Asymptotes:**
- Vertical asymptotes are usually found where the denominator of the rational function is equal to zero (i.e., \( Q(x) = 0 \)), provided that the numerator \( P(x) \) is not also equal to zero at that point. If both \( P(x) \) and \( Q(x) \) are zero, it may indicate a removable discontinuity instead.

### Steps to Find Vertical Asymptotes of a Rational Function:

1. **Identify the Rational Function:**
- Start with a rational function of the form \( f(x) = \frac{P(x)}{Q(x)} \).

2. **Set the Denominator to Zero:**
- To find potential vertical asymptotes, solve the equation:
\[
Q(x) = 0.
\]
- The solutions to this equation give the values of \( x \) that may correspond to vertical asymptotes.

3. **Check the Numerator:**
- For each value \( x = a \) obtained from \( Q(x) = 0 \), check whether \( P(a) \) is also zero:
- If \( P(a) \neq 0 \), then there is a vertical asymptote at \( x = a \).
- If \( P(a) = 0 \), the function may have a removable discontinuity instead.

4. **Confirm the Asymptotic Behavior:**
- Although not always necessary, it’s useful to confirm the asymptotic behavior by evaluating the limits:
- Compute \( \lim_{x \to a^-} f(x) \) and \( \lim_{x \to a^+} f(x) \) to see if one limit approaches \( +\infty \) and the other approaches \( -\infty \) (or vice versa), confirming a vertical asymptote.

### Example:

Consider the rational function:
\[
f(x) = \frac{x^2 - 4}{x^2 - 1}.
\]

1. **Identify the Rational Function:**
- The function is \( f(x) = \frac{x^2 - 4}{x^2 - 1} \).

2. **Set the Denominator to Zero:**
- Solve for \( x \):
\[
x^2 - 1 = 0 \implies x^2 = 1 \implies x = \pm 1.
\]
- This gives potential vertical asymptotes at \( x = 1 \) and \( x = -1 \).

3. **Check the Numerator:**
- Evaluate \( P(x) = x^2 - 4 \) at the critical points:
- For \( x = 1 \):
\[
P(1) = 1^2 - 4 = -3 \neq 0.
\]
- For \( x = -1 \):
\[
P(-1) = (-1)^2 - 4 = -3 \neq 0.
\]
- Since \( P(1) \) and \( P(-1) \) are both non-zero, both \( x = 1 \) and \( x = -1 \) are vertical asymptotes.

4. **Confirm the Asymptotic Behavior:**
- Compute the limits to confirm:
- For \( x \to 1 \):
\[
\lim_{x \to 1^-} f(x) \to +\infty \quad \text{and} \quad \lim_{x \to 1^+} f(x) \to -\infty.
\]
- For \( x \to -1 \):
\[
\lim_{x \to -1^-} f(x) \to +\infty \quad \text{and} \quad \lim_{x \to -1^+} f(x) \to -\infty.
\]
- These results confirm that there are vertical asymptotes at both \( x = 1 \) and \( x = -1 \).

### Summary:

Finding vertical asymptotes of a rational function is a fundamental concept in AP Calculus AB. By setting the denominator equal to zero and ensuring that the numerator does not equal zero at those points, students can identify critical points that indicate where the function may behave asymptotically. This understanding is essential for analyzing the overall behavior and graphing of rational functions in calculus.

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Nick Perich
Norristown Area High School
Norristown Area School District
Norristown, Pa

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