To evaluate the limit \( \lim_{h \to 0} \frac{\sin(h)}{3h} \), we can use a known trigonometric limit identity and apply it to the given expression.

### Step-by-Step Solution:
1. Recall the fundamental limit:
   \[
   \lim_{h \to 0} \frac{\sin(h)}{h} = 1
   \]
2. Rewrite the given limit to match the form of this known limit:
   \[
   \lim_{h \to 0} \frac{\sin(h)}{3h} = \frac{1}{3} \lim_{h \to 0} \frac{\sin(h)}{h}
   \]
3. Using the known result:
   \[
   \lim_{h \to 0} \frac{\sin(h)}{h} = 1
   \]
   we get:
   \[
   \frac{1}{3} \times 1 = \frac{1}{3}
   \]

### Final Answer:
\[
\lim_{h \to 0} \frac{\sin(h)}{3h} = \frac{1}{3}
\]

Would you like more details or have any questions?

Here are 5 related questions:
1. What is the limit of \( \lim_{h \to 0} \frac{\sin(2h)}{h} \)?
2. How do you apply L'Hopital's Rule to evaluate limits?
3. What are other key trigonometric limits useful for calculus?
4. What is the significance of the small-angle approximation for sine?
5. How do you find limits involving \(\sin(x)\) and other trigonometric functions?

**Tip:** Knowing key limits like \( \lim_{x \to 0} \frac{\sin(x)}{x} = 1 \) is essential for solving many calculus problems involving trigonometric functions.

Discover how to evaluate the limit lim h→0 sin(h)/3h with a detailed step-by-step solution. This problem involves key calculus concepts, including the fundamental trigonometric limit. Suitable for students in Grades 11-12.

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