The problem provided asks for the value of the following limit:

$\lim_{x \to 0} \frac{\sin(5x) - \sin(3x)}{\sin(x)}$

Solution:

We can approach this limit using trigonometric identities and L'Hôpital's Rule.

1. Difference of Sines Identity: We use the identity: $\sin A - \sin B = 2 \cos\left(\frac{A+B}{2}\right) \sin\left(\frac{A-B}{2}\right)$ Here, let $A = 5x$ and $B = 3x$, so:

   = 2 \cos(4x) \sin(x)$$

2. Substitute into the Limit Expression: Substitute the above expression into the limit: $\lim_{x \to 0} \frac{\sin(5x) - \sin(3x)}{\sin(x)} = \lim_{x \to 0} \frac{2 \cos(4x) \sin(x)}{\sin(x)}$

3. Simplify the Expression: Simplifying the expression by canceling $\sin(x)$: $\lim_{x \to 0} 2 \cos(4x)$

4. Evaluate the Limit: As $x \to 0$, $\cos(4x) \to \cos(0) = 1$, so the limit becomes: $\lim_{x \to 0} 2 \cos(4x) = 2 \times 1 = 2$

Final Answer:

$\lim_{x \to 0} \frac{\sin(5x) - \sin(3x)}{\sin(x)} = 2$

Would you like further details or have any other questions?

Related Questions:

1. What is L'Hôpital's Rule, and how is it applied to limits?
2. How do you derive the difference of sines identity used here?
3. Can you explain the behavior of trigonometric functions as $x$ approaches 0?
4. What are other common trigonometric limits?
5. How would the limit change if the numerator was $\sin(5x) - \sin(2x)$?
6. What happens if you apply Taylor series expansions instead?
7. How can you use the small angle approximation in similar limit problems?
8. What other types of limits can be solved using trigonometric identities?

Tip: For limit problems involving trigonometric functions, knowing trigonometric identities and the behavior of these functions near zero is crucial for simplifying and solving the expressions effectively.