AP Calculus AB 1.13 Determining Points of Removable Discontinuity in a Rational Function

In AP Calculus AB, topic 1.13 focuses on determining points of removable discontinuity in rational functions. Understanding removable discontinuities is essential for students as it helps them analyze the behavior of functions, particularly in the context of limits and continuity.

### Key Concepts:

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

2. **Understanding Discontinuities:**
- Discontinuities in rational functions can occur when the denominator \( Q(x) \) is equal to zero. These discontinuities can be classified as removable or non-removable.

3. **Removable Discontinuity:**
- A removable discontinuity occurs when a factor in the denominator cancels with a corresponding factor in the numerator. This means the function can be redefined to make it continuous at that point.

### Steps to Identify Points of Removable Discontinuity:

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

2. **Factor the Numerator and Denominator:**
- Factor both the numerator \( P(x) \) and the denominator \( Q(x) \) completely to find common factors.

3. **Set the Denominator Equal to Zero:**
- Find the values of \( x \) that make the denominator zero by solving the equation \( Q(x) = 0 \). These are potential points of discontinuity.

4. **Check for Common Factors:**
- Look for any common factors between the numerator and denominator. If a factor cancels, this indicates a removable discontinuity at that point.

5. **Determine the Value of the Function:**
- For a removable discontinuity at \( x = c \), you can redefine the function by removing the factor causing the discontinuity and substituting \( x = c \) back into the simplified function to find the limit:
\[
\lim_{x \to c} f(x) = \text{value at } x = c.
\]

### Example:

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

1. **Factor the Function:**
- Factor the numerator:
\[
f(x) = \frac{(x - 2)(x + 2)}{x - 2}
\]

2. **Set the Denominator Equal to Zero:**
- The denominator \( x - 2 = 0 \) gives \( x = 2 \). This is a potential point of discontinuity.

3. **Check for Common Factors:**
- The factor \( (x - 2) \) cancels from both the numerator and the denominator. Therefore, there is a removable discontinuity at \( x = 2 \).

4. **Redefine the Function:**
- The simplified function is:
\[
f(x) = x + 2 \text{ for } x \neq 2.
\]
- To find the limit as \( x \) approaches 2:
\[
\lim_{x \to 2} f(x) = 2 + 2 = 4.
\]

5. **Conclusion:**
- The point \( x = 2 \) is a removable discontinuity. The function can be redefined at this point (for example, \( f(2) = 4 \)) to make it continuous.

### Summary:

Understanding how to determine points of removable discontinuity in rational functions is essential for students in AP Calculus AB. By following the steps of factoring, identifying common factors, and analyzing limits, students can effectively find and address discontinuities in rational functions. This knowledge is crucial for understanding limits, continuity, and the overall behavior of functions in calculus.
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Nick Perich
Norristown Area High School
Norristown Area School District
Norristown, Pa

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