https://file.aiquickdraw.com/mathbot/file/1725387719946wkyvwkr8.jpg

The limit you provided is:

\[
\lim_{x \to 0} \frac{(\sin 3x)^2}{x^2 \cos x}
\]

### Steps to Solve:

1. **Expression Simplification**: Start by expressing \(\sin 3x\) in a more convenient form near \(x = 0\).

   \[
   \sin 3x \approx 3x \quad \text{as} \quad x \to 0
   \]

   So,

   \[
   (\sin 3x)^2 \approx (3x)^2 = 9x^2
   \]

2. **Substitute in the Limit**:

   Substitute this approximation into the original limit expression:

   \[
   \lim_{x \to 0} \frac{9x^2}{x^2 \cos x}
   \]

3. **Simplification**:

   Simplify the expression by canceling \(x^2\) in the numerator and denominator:

   \[
   \lim_{x \to 0} \frac{9}{\cos x}
   \]

4. **Evaluate the Limit**:

   As \(x \to 0\), \(\cos x \to 1\). Therefore:

   \[
   \lim_{x \to 0} \frac{9}{\cos x} = \frac{9}{1} = 9
   \]

### Final Answer:

\[
\lim_{x \to 0} \frac{(\sin 3x)^2}{x^2 \cos x} = 9
\]

Would you like more details or have any other questions?

---

Here are 5 related questions to further your understanding:

1. How would the limit change if the expression were \(\lim_{x \to 0} \frac{(\sin 5x)^2}{x^2 \cos x}\)?
2. What happens if \(\cos x\) were replaced with \(\cos(2x)\) in the denominator?
3. How does L'Hôpital's Rule apply to solving limits involving indeterminate forms like \(\frac{0}{0}\)?
4. What is the approximation for \(\sin x\) near \(x = 0\), and why is it useful?
5. Can you apply the small-angle approximation to \(\cos x\) to verify the solution?

**Tip:** Small-angle approximations are useful in simplifying trigonometric functions near 0, especially in limit problems.

Evaluate the limit lim_{x→0} (sin(3x))^2 / (x^2 cos(x)).

Learn how to evaluate the limit lim_{x→0} (sin(3x))^2 / (x^2 cos(x)) with a detailed step-by-step solution. This problem covers key concepts in limits, trigonometry, and the small-angle approximation, making it suitable for students in Grades 11-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation