The expression provided in the image is:

$\lim_{x \to 3} \frac{2 \sin(x^{10})}{4x^{10}}$

Let's simplify and evaluate this limit.

Step 1: Simplify the Expression

The expression inside the limit can be simplified as follows:

$\lim_{x \to 3} \frac{2 \sin(x^{10})}{4x^{10}} = \lim_{x \to 3} \frac{\sin(x^{10})}{2x^{10}}$

Step 2: Substitute $x = 3$

Substituting $x = 3$ directly into the expression:

$\frac{\sin(3^{10})}{2 \times 3^{10}} = \frac{\sin(59049)}{2 \times 59049}$

Step 3: Evaluate the Sine Function

The value of $\sin(59049)$ would be a specific number, but because it's inside a sine function, it will be within the range $[-1, 1]$.

So, the limit becomes:

$\lim_{x \to 3} \frac{\sin(59049)}{2 \times 59049}$

This fraction is quite small since $2 \times 59049$ is a large number, and $\sin(59049)$ is bounded between $-1$ and $1$.

Final Answer:

The limit is:

$\lim_{x \to 3} \frac{\sin(3^{10})}{2 \times 3^{10}} \approx \frac{\sin(59049)}{118098}$

This is a very small number because the sine function is divided by a large denominator. The exact value depends on the sine function but will be quite close to zero.

Would you like more details on any specific part, or do you have any questions?

Here are five related questions:

1. How does the limit behave as $x$ approaches values near zero?
2. What if the expression were $\lim_{x \to 0} \frac{\sin(x^{10})}{x^{10}}$?
3. How does changing the exponent from 10 to 2 affect the limit?
4. What role does the periodic nature of the sine function play in determining limits?
5. Can you explain the significance of L'Hôpital's Rule in solving limits involving sine functions?

Tip: When evaluating limits involving trigonometric functions, consider the behavior of the function and the input value's magnitude.