https://file.aiquickdraw.com/mathbot/file/17271639089789suo62tc.jpg

It seems that the OCR was unable to extract the mathematical expression from the image. However, from the visual inspection, the problem appears to be:

\[
\lim_{{x \to \frac{1}{2}}} \frac{2x^2 - 5x + 2}{\sin(4x - 2)}
\]

Let's solve this limit step-by-step.

### Step 1: Evaluate the limit directly
Substitute \( x = \frac{1}{2} \) into both the numerator and the denominator:

1. **Numerator**: \( 2x^2 - 5x + 2 \)
   \[
   2\left(\frac{1}{2}\right)^2 - 5\left(\frac{1}{2}\right) + 2 = 2\left(\frac{1}{4}\right) - \frac{5}{2} + 2 = \frac{1}{2} - \frac{5}{2} + 2 = 0
   \]
2. **Denominator**: \( \sin(4x - 2) \)
   \[
   \sin(4\left(\frac{1}{2}\right) - 2) = \sin(2 - 2) = \sin(0) = 0
   \]

Since both the numerator and the denominator are 0, this is an indeterminate form \( \frac{0}{0} \). We need to use **L'Hopital's Rule**.

### Step 2: Apply L'Hopital's Rule
L'Hopital's Rule states that for limits of the form \( \frac{0}{0} \), we can differentiate the numerator and the denominator separately and then take the limit again.

- **Differentiate the numerator**: \( 2x^2 - 5x + 2 \)
   \[
   \frac{d}{dx}(2x^2 - 5x + 2) = 4x - 5
   \]

- **Differentiate the denominator**: \( \sin(4x - 2) \)
   \[
   \frac{d}{dx}(\sin(4x - 2)) = 4 \cos(4x - 2)
   \]

### Step 3: Evaluate the new limit
Now, take the limit of the new expression:
\[
\lim_{{x \to \frac{1}{2}}} \frac{4x - 5}{4 \cos(4x - 2)}
\]

Substitute \( x = \frac{1}{2} \) into the new expression:
1. **Numerator**: \( 4x - 5 \)
   \[
   4\left(\frac{1}{2}\right) - 5 = 2 - 5 = -3
   \]

2. **Denominator**: \( 4 \cos(4x - 2) \)
   \[
   4 \cos(4\left(\frac{1}{2}\right) - 2) = 4 \cos(2 - 2) = 4 \cos(0) = 4 \times 1 = 4
   \]

So the limit is:
\[
\frac{-3}{4}
\]

Thus, the final answer is:
\[
\lim_{{x \to \frac{1}{2}}} \frac{2x^2 - 5x + 2}{\sin(4x - 2)} = \frac{-3}{4}
\]

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

Here are some related questions you might find helpful:
1. What is the significance of L'Hopital's Rule in solving limits?
2. How do you differentiate trigonometric functions like sine and cosine?
3. Can you explain how indeterminate forms work in calculus?
4. What happens if L'Hopital's Rule needs to be applied multiple times?
5. How would this limit change if the function involved a different trigonometric function, like cosine?

**Tip**: Always check for indeterminate forms \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \) before applying L'Hopital's Rule.

Learn how to solve a limit problem using L'Hopital's Rule. The problem involves the expression lim_{x → 1/2} (2x^2 - 5x + 2)/sin(4x - 2), where both the numerator and denominator yield 0. The solution includes differentiating and applying trigonometric identities to reach the final result of -3/4. Suitable for advanced high school and college-level students.

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