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The given problem asks to find the value of the limit:

\[
\lim_{x \to \infty} \left( \frac{1}{3} x \cdot \cos\left(\frac{3}{x}\right) \cdot \sin\left(\frac{1}{5x}\right) \right).
\]

### Step-by-Step Solution

1. **Analyze the components of the limit:**

   - As \( x \to \infty \), the angle \( \frac{3}{x} \to 0 \). The function \( \cos\left(\frac{3}{x}\right) \) approaches \( \cos(0) = 1 \).
   - Similarly, the angle \( \frac{1}{5x} \to 0 \). The function \( \sin\left(\frac{1}{5x}\right) \) behaves like \( \frac{1}{5x} \) for small values of \( \frac{1}{5x} \).

2. **Approximation for small angles:**

   When \( x \to \infty \),
   - \( \cos\left(\frac{3}{x}\right) \approx 1 \),
   - \( \sin\left(\frac{1}{5x}\right) \approx \frac{1}{5x} \).

   Therefore, the expression inside the limit can be approximated as:
   \[
   \frac{1}{3} x \cdot 1 \cdot \frac{1}{5x} = \frac{1}{3} \cdot \frac{1}{5} = \frac{1}{15}.
   \]

3. **Conclusion:**
   The value of the limit is \( \frac{1}{15} \).

Thus, the correct answer is \( \boxed{A} \).

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Do you want further details on any step or have any questions?

Here are 5 related questions to explore:
1. How does the small-angle approximation for sine and cosine work?
2. What is the behavior of trigonometric functions as their angles approach zero?
3. Can L'Hôpital's Rule be used for this problem? If so, how?
4. How would the result change if the cosine or sine function had a different coefficient?
5. What is the significance of limits in calculus, particularly when approaching infinity?

**Tip:** When dealing with limits that involve trigonometric functions and small angles, it's often useful to apply approximations like \( \sin(\theta) \approx \theta \) for very small \( \theta \).

Find the limit of the following expression as x approaches infinity: (1/3) * x * cos(3/x) * sin(1/5x).

This problem involves finding the limit of the expression (1/3) * x * cos(3/x) * sin(1/5x) as x approaches infinity. The solution explores key concepts in calculus and trigonometry, including the small-angle approximation and limit of a product of functions. Suitable for students in Grades 10-12.

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