https://file.aiquickdraw.com/mathbot/file/17257062851592mewjsra.jpg

It seems that the text extraction from the image was not successful. I will analyze the expression visually instead.

The limit provided in the image is:

\[
\lim_{{x \to 0}} \frac{6x - \sin(2x)}{2x + 3 \sin(4x)}
\]

### Solving the limit:

1. **Direct substitution**: Substituting \(x = 0\) into the expression:
   - \(\sin(2x) = \sin(0) = 0\)
   - \(\sin(4x) = \sin(0) = 0\)
   
   The expression becomes:

   \[
   \frac{6(0) - \sin(0)}{2(0) + 3 \sin(0)} = \frac{0}{0}
   \]
   
   This results in an indeterminate form \( \frac{0}{0} \), so we will apply **L'Hôpital's Rule**, which is used when limits result in \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \).

2. **Applying L'Hôpital's Rule**: Differentiate the numerator and denominator with respect to \(x\):
   - Numerator: \( \frac{d}{dx} (6x - \sin(2x)) = 6 - 2 \cos(2x) \)
   - Denominator: \( \frac{d}{dx} (2x + 3 \sin(4x)) = 2 + 12 \cos(4x) \)

3. **Evaluate the limit again** by substituting \(x = 0\):
   - Numerator: \( 6 - 2 \cos(0) = 6 - 2(1) = 4 \)
   - Denominator: \( 2 + 12 \cos(0) = 2 + 12(1) = 14 \)

   So, the limit becomes:

   \[
   \lim_{{x \to 0}} \frac{4}{14} = \frac{2}{7}
   \]

Thus, the solution is:

\[
\boxed{\frac{2}{7}}
\]

Would you like any further explanation on L'Hôpital's Rule or related concepts?

Here are five related questions to deepen your understanding:
1. How do you determine when to apply L'Hôpital's Rule?
2. What are common indeterminate forms in limits?
3. How do you differentiate trigonometric functions like \(\sin(x)\) and \(\cos(x)\)?
4. Can you apply L'Hôpital's Rule multiple times if the indeterminate form persists?
5. What other methods can be used to solve limits when L'Hôpital's Rule is not applicable?

**Tip**: Always check if direct substitution leads to an indeterminate form before applying L'Hôpital's Rule.

Evaluate the limit: lim(x -> 0) (6x - sin(2x)) / (2x + 3sin(4x))

Learn how to solve the limit lim(x -> 0) (6x - sin(2x)) / (2x + 3sin(4x)) using L'Hôpital's Rule. This problem covers key calculus concepts such as limits, derivatives, and trigonometric functions. Suitable for advanced high school students.

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