Problem 47 - AP Calculus with 8 Practice Tests 15th

Question

\(\lim _{x \rightarrow 0} \frac{\sin 3 x}{\sin 4 x}\) is (A) 1 (B) \(\frac{4}{3}\) (C) \(\frac{3}{4}\) (D) 0

Step-by-Step Solution

Answer

(C) \(\frac{3}{4}\).

Step 1: Recognize the Indeterminate Form

Substitute \(x = 0\) into \(\frac{\sin 3x}{\sin 4x}\). It results in \(\frac{0}{0}\), indicating the need for algebraic manipulation or a limit property.

Step 2: Apply Trigonometric Limit Property

Recall that \(\lim_{x \to 0} \frac{\sin kx}{kx} = 1\) for any constant \(k\). Thus, we write the expression as \(\frac{\sin 3x}{3x} \times \frac{4x}{\sin 4x} \times \frac{3}{4}\).

Step 3: Evaluate Individual Limits

Evaluate \(\lim_{x \to 0} \frac{\sin 3x}{3x} = 1\) and \(\lim_{x \to 0} \frac{4x}{\sin 4x} = 1\).

Step 4: Compute the Overall Limit

Multiply the results from Step 3: \(1 \cdot 1 \cdot \frac{3}{4} = \frac{3}{4}\). Therefore, the limit is \(\frac{3}{4}\).

Key Concepts

Indeterminate Forms

When evaluating limits, especially in calculus, you might come across expressions where directly substituting the variable gives you a form like \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \). These are known as indeterminate forms. They don't tell us about the behavior of the function as the variable approaches a particular point. Instead, they signal the need for deeper analysis or more refined techniques to understand the limit's value.

For example, when we tried to find the limit \( \lim_{x \rightarrow 0} \frac{\sin 3x}{\sin 4x} \), substituting \(x = 0\) directly into the expression yielded \( \frac{0}{0} \). This indicates an indeterminate form and suggests that a straightforward evaluation wouldn't work. In such cases, we often have to manipulate the expression or use specific limit properties to evaluate it correctly.

Approaches to resolve indeterminate forms often include algebraic manipulation, applying L'Hôpital's rule, or using standard limit properties. These methods help us bypass the ambiguities and ascertain the true behavior of the function as it approaches a certain point.

Trigonometric Limits

Trigonometric limits play a crucial role in solving problems involving trigonometric functions. One of the most important fundamental limits in this area is \( \lim_{x \to 0} \frac{\sin kx}{kx} = 1 \). This property is invaluable because it allows us to work around indeterminate forms involving sine functions.

In our example, \( \lim_{x \rightarrow 0} \frac{\sin 3x}{\sin 4x} \), we faced the \( \frac{0}{0} \) indeterminate form. By using the trigonometric limit property, we can rewrite the expression as \( \frac{\sin 3x}{3x} \cdot \frac{4x}{\sin 4x} \cdot \frac{3}{4} \). This transformation introduces manageable parts that align with our key trigonometric limit property.

Breaking it down further:

• The expression \( \frac{\sin 3x}{3x} \) approaches 1 as \( x \rightarrow 0 \).
• Similarly, \( \frac{4x}{\sin 4x} \) also approaches 1 under these conditions.

Through this series of conversions and simplifications, trigonometric limits help us unveil the behavior of functions where direct substitution fails.

Limit Properties

Limit properties are foundational tools in calculus. They help simplify and evaluate limits when direct substitution isn't feasible due to indeterminate forms. These properties provide rules that govern how limits can be combined, manipulated, and understood.

Key limit properties include:

• If \( \lim_{x \to a} f(x) = L \) and \( \lim_{x \to a} g(x) = M \), then \( \lim_{x \to a} (f(x) \cdot g(x)) = L \cdot M \).
• The sum of limits follows: \( \lim_{x \to a} (f(x) + g(x)) = L + M \).
• Similarly, \( \lim_{x \to a} \left( \frac{f(x)}{g(x)} \right) = \frac{L}{M} \), given that \( M eq 0 \).

These properties were exploited in the example exercise to evaluate the overall limit.

After expressing the problem using trigonometric limits and recognizing individual limit behaviors as 1, we applied the product property. We combined them to find the overarching limit: \( 1 \cdot 1 \cdot \frac{3}{4} = \frac{3}{4} \). Through these properties, complex expressions become tractable, guiding us to a clear and accurate limit calculation.