I will examine the image you uploaded to extract the mathematical content and provide an explanation or solution. Let me analyze it first.The image you uploaded contains a mathematical problem related to calculus, specifically focusing on limits, derivatives, and the application of rules like L'Hôpital's Rule.

Extracted Problem:

The question appears to be related to evaluating a limit. From the text in the image, it seems to ask for the evaluation of the following limit:

$\lim_{x \to 0} \frac{\sin(4x)}{\sin(6x)}$

Solution:

To evaluate the limit $\lim_{x \to 0} \frac{\sin(4x)}{\sin(6x)}$, we can use the following trigonometric limit identity:

$\lim_{x \to 0} \frac{\sin(kx)}{kx} = 1$

First, let's rewrite the expression:

$\lim_{x \to 0} \frac{\sin(4x)}{\sin(6x)} = \lim_{x \to 0} \frac{\sin(4x)}{4x} \cdot \frac{6x}{\sin(6x)} \cdot \frac{4x}{6x}$

Now, apply the limit identity:

$= \lim_{x \to 0} \left(\frac{\sin(4x)}{4x}\right) \cdot \lim_{x \to 0} \left(\frac{6x}{\sin(6x)}\right) \cdot \frac{4}{6}$

Since both $\lim_{x \to 0} \frac{\sin(4x)}{4x} = 1$ and $\lim_{x \to 0} \frac{6x}{\sin(6x)} = 1$, the limit simplifies to:

$= 1 \cdot 1 \cdot \frac{4}{6} = \frac{4}{6} = \frac{2}{3}$

Final Answer:

$\lim_{x \to 0} \frac{\sin(4x)}{\sin(6x)} = \frac{2}{3}$

Would you like more details or have any questions? Here are some additional problems and tips based on the solution:

1. How can L'Hôpital's Rule be applied to solve similar limit problems?
2. What are the limits of other trigonometric functions as $x$ approaches 0?
3. How does the small-angle approximation help in solving limits involving trigonometric functions?
4. Can you generalize the result for $\lim_{x \to 0} \frac{\sin(ax)}{\sin(bx)}$?
5. What is the importance of knowing these limit identities in calculus?

Tip: When solving limit problems involving trigonometric functions, always check if the small-angle approximation or known limits can simplify the problem.