It looks like you're asking for the limit of the function involving \( x^2 \) and \( \cos(3x) \) as \( x \to 0 \). Here's the expression I'm interpreting:

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

### Step-by-step Solution:

1. The function you're dealing with is \( f(x) = x^2 \cos(3x) \).
2. As \( x \to 0 \), let's examine each part:
   - \( \cos(3x) \) tends to \( \cos(0) = 1 \).
   - \( x^2 \) tends to 0 since the square of any small number is even smaller.
3. So, the limit becomes:
   \[
   \lim_{x \to 0} x^2 \cos(3x) = \lim_{x \to 0} x^2 \cdot 1 = 0.
   \]

### Final Answer:
\[
\lim_{x \to 0} x^2 \cos(3x) = 0.
\]

Would you like further details or clarifications?

Here are some related questions you might consider:
1. What is the behavior of \( x^2 \cos(3x) \) as \( x \to \infty \)?
2. How does the cosine function influence limits with rapidly decaying functions like \( x^2 \)?
3. What happens to the limit if we change \( x^2 \) to \( x^n \), where \( n > 2 \)?
4. Can L'Hôpital's Rule be applied to this type of limit problem?
5. What is the limit of \( \cos(3x) \) alone as \( x \to 0 \)?

**Tip:** When finding limits of products of functions, always check if one part of the product tends to zero faster than the others.

Learn how to solve the limit of the function x^2 cos(3x) as x approaches 0. This problem involves key concepts from limits and trigonometric functions, including applying the limit product rule. Suitable for high school students in Grades 11-12.

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