AP Calculus AB 4.7 Using L'Hôpital's Rule to Find the Limit of a Rational Function ( sin(3x) )

### AP Calculus AB 4.7: Using L'Hôpital's Rule to Find the Limit of a Rational Function (sin(3x))

**Overview**: This section focuses on applying L'Hôpital's Rule to evaluate limits involving the sine function, particularly \( \sin(3x) \), within the context of rational functions. Understanding how to analyze these limits, especially when they lead to indeterminate forms, is crucial for students in AP Calculus AB.

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### **Key Concepts**

#### **1. Sine Function**
- The sine function, \( \sin(x) \), is periodic and oscillates between -1 and 1 for all real values of \( x \).
- The behavior of \( \sin(3x) \) compresses the sine wave, causing it to complete three cycles within the interval of \( 0 \) to \( 2\pi \).

#### **2. Indeterminate Forms**
- Limits involving \( \sin(3x) \) can often lead to indeterminate forms, such as:
- \( \frac{0}{0} \)
- \( \frac{\infty}{\infty} \)

For example, evaluating:
\[
\lim_{x \to 0} \frac{\sin(3x)}{x}
\]
yields the indeterminate form \( \frac{0}{0} \).

#### **3. L'Hôpital's Rule**
- L'Hôpital's Rule can be applied to resolve limits of the form \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \):
\[
\lim_{x \to c} \frac{f(x)}{g(x)} = \lim_{x \to c} \frac{f'(x)}{g'(x)}
\]
provided the limit on the right exists.

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### **Applying L'Hôpital's Rule with sin(3x)**

#### **Step-by-Step Process**:

1. **Evaluate the Limit**:
- Substitute the value into the limit to check for indeterminate forms.

2. **Differentiate the Numerator and Denominator**:
- If the limit results in an indeterminate form, differentiate the numerator (which may include \( \sin(3x) \)) and the denominator separately.

3. **Re-evaluate the Limit**:
- Substitute the value again into the new fraction \( \frac{f'(x)}{g'(x)} \).
- If the new limit still results in an indeterminate form, you can apply L'Hôpital's Rule again.

4. **Conclude the Limit**:
- If the limit can be evaluated without yielding another indeterminate form, simplify and conclude the limit.

---

### **Example Problem**:

**Evaluate the Limit**:
\[
\lim_{x \to 0} \frac{\sin(3x)}{x}
\]

1. **Direct Substitution**:
\[
\frac{\sin(3 \cdot 0)}{0} = \frac{0}{0} \quad \text{(Indeterminate)}
\]

2. **Differentiate**:
- Numerator: \( f(x) = \sin(3x) \) → \( f'(x) = 3\cos(3x) \)
- Denominator: \( g(x) = x \) → \( g'(x) = 1 \)

3. **Re-evaluate**:
\[
\lim_{x \to 0} \frac{3\cos(3x)}{1} = 3\cos(0) = 3 \cdot 1 = 3
\]

4. **Conclusion**:
- Therefore, \( \lim_{x \to 0} \frac{\sin(3x)}{x} = 3 \).

---

### **Further Considerations**

- **Multiple Applications**: Students may need to apply L'Hôpital's Rule multiple times, especially for more complex functions or limits.

- **General Limit Forms**: Students should also become familiar with the general limit property:
\[
\lim_{x \to 0} \frac{\sin(kx)}{x} = k
\]
which can be derived from applying L'Hôpital's Rule or using Taylor series expansion.

- **Graphical Interpretation**: Understanding the behavior of the sine function near zero can provide insight into the limit evaluation process.

---

### **Conclusion**:
Using L'Hôpital's Rule to find limits involving \( \sin(3x) \) provides students with essential skills for evaluating limits that lead to indeterminate forms. Mastery of this technique is crucial for success in AP Calculus AB and establishes a solid foundation for further studies in calculus and analysis.

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

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