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.