https://file.aiquickdraw.com/mathbot/file/17264481711564itaj0hc.jpg

The problem asks you to find the limit of the following expression:

\[
\lim_{x \to 0} \frac{\sin^2(6x)}{\sin^2(12x)}
\]

### Step-by-Step Solution:

1. **Rewrite the expression:**

   Use the fact that \(\sin^2(6x)\) and \(\sin^2(12x)\) can be separated into two squared sine functions:

   \[
   \lim_{x \to 0} \left( \frac{\sin(6x)}{\sin(12x)} \right)^2
   \]

2. **Apply the standard trigonometric limit**: 
   
   One useful limit to remember is:
   \[
   \lim_{x \to 0} \frac{\sin(kx)}{kx} = 1 \quad \text{for any constant } k
   \]
   So we can multiply both the numerator and the denominator by \(x\) to use this rule:
   
   \[
   \lim_{x \to 0} \left( \frac{\frac{\sin(6x)}{6x}}{\frac{\sin(12x)}{12x}} \times \frac{12x}{6x} \right)^2
   \]
   
   Since \(\lim_{x \to 0} \frac{\sin(kx)}{kx} = 1\), the fractions involving sine tend to 1. Thus, the limit simplifies to:
   
   \[
   \left( \frac{12}{6} \right)^2 = 2^2 = 4
   \]

### Final Answer:
\[
\lim_{x \to 0} \frac{\sin^2(6x)}{\sin^2(12x)} = 4
\]

Would you like more details on any step, or do you have any questions?

---

Here are five related questions to extend your understanding:

1. What is the limit of \(\lim_{x \to 0} \frac{\sin(3x)}{\sin(6x)}\)?
2. How do you compute \(\lim_{x \to 0} \frac{\sin(x)}{x}\) using L'Hôpital's Rule?
3. What happens if we have \(\lim_{x \to 0} \frac{\sin(kx)}{\sin(mx)}\) for arbitrary constants \(k\) and \(m\)?
4. Can the method of multiplying by \(x\) be applied to other trigonometric functions such as \(\tan(x)\)?
5. How do we evaluate limits involving trigonometric functions and higher powers, like \(\sin^3(x)\)?

### Tip:
When working with trigonometric limits, always look for opportunities to apply standard limits like \(\lim_{x \to 0} \frac{\sin(x)}{x} = 1\) to simplify the expressions.

Find the limit: lim (x -> 0) [sin^2(6x) / sin^2(12x)]

Learn how to solve the limit lim (x -> 0) [sin^2(6x) / sin^2(12x)] with a step-by-step solution, applying standard trigonometric limits. This problem is suitable for high school students learning about limits and trigonometry.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation