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To solve the given limit:

\[
\lim_{x \to 0} \left( \cos(2x) \cdot \frac{1}{\sin^2(3x)} \right)
\]

### Step 1: Simplify the expression
First, notice that the expression inside the limit can be written as:

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

### Step 2: Evaluate each component at the limit
We know that as \(x \to 0\):
- \(\cos(2x)\) approaches \(\cos(0) = 1\).
- \(\sin(3x)\) approaches \(\sin(0) = 0\).

However, this creates an indeterminate form \( \frac{1}{0^2} \), which tends to infinity. Therefore, we need to use the small-angle approximation.

### Step 3: Apply small-angle approximations
For small values of \( x \), the following approximations hold:
- \(\cos(2x) \approx 1 - \frac{(2x)^2}{2} = 1 - 2x^2\)
- \(\sin(3x) \approx 3x\)

Thus, we can rewrite the limit as:

\[
\lim_{x \to 0} \frac{1 - 2x^2}{(3x)^2}
\]

### Step 4: Simplify the expression further
Simplifying the expression inside the limit:

\[
\frac{1 - 2x^2}{9x^2}
\]

This can be separated as:

\[
\lim_{x \to 0} \left(\frac{1}{9x^2} - \frac{2x^2}{9x^2}\right) = \lim_{x \to 0} \left(\frac{1}{9x^2} - \frac{2}{9}\right)
\]

As \( x \to 0 \), the first term \( \frac{1}{9x^2} \) tends to infinity, while the second term \( \frac{2}{9} \) is a constant.

### Step 5: Conclusion
Since \( \frac{1}{9x^2} \) dominates as \( x \) approaches 0, the entire limit tends towards infinity:

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

Thus, the limit is **infinite**.

---

Would you like further details or have any questions? Here are some related questions:

1. How do you apply the small-angle approximation to other trigonometric functions?
2. What are the conditions for using L'Hôpital's Rule in limits?
3. How can limits approaching infinity be interpreted in real-world problems?
4. What are some common indeterminate forms, and how are they resolved?
5. How would the result change if we evaluated the limit of a different function, like \(\frac{\sin(2x)}{\sin^2(3x)}\)?

**Tip:** Always check if a limit yields an indeterminate form; it can guide you toward the appropriate method of resolution.

Learn how to solve the limit \(\lim_{x \to 0} \frac{\cos(2x)}{\sin^2(3x)}\) with detailed step-by-step instructions. This problem involves trigonometric functions, small-angle approximation, and understanding limits. Suitable for advanced high school students and those studying calculus.

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