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The text extraction from the image was not successful again, so I'll analyze the limit expression directly from the image.

The limit is given as:

\[
\lim_{{x \to \frac{1}{2}}} \left[ \frac{2x^3 - 1}{(8x^2 + 2)(5x - 5)} - \frac{2 - 5x}{x^4 - 5x + 1} \right]
\]

### Approach:

1. **Substitute \(x = \frac{1}{2}\)** into both parts of the expression.
   
   - In the first term: 
     \[
     \frac{2\left(\frac{1}{2}\right)^3 - 1}{(8\left(\frac{1}{2}\right)^2 + 2)(5\left(\frac{1}{2}\right) - 5)}
     = \frac{2\left(\frac{1}{8}\right) - 1}{(8\left(\frac{1}{4}\right) + 2)(5\left(\frac{1}{2}\right) - 5)}
     = \frac{\frac{1}{4} - 1}{(2 + 2)(\frac{5}{2} - 5)}
     = \frac{-\frac{3}{4}}{(4)(-\frac{5}{2})}
     = \frac{-\frac{3}{4}}{-10} = \frac{3}{40}
     \]
   
   - In the second term:
     \[
     \frac{2 - 5\left(\frac{1}{2}\right)}{\left(\frac{1}{2}\right)^4 - 5\left(\frac{1}{2}\right) + 1}
     = \frac{2 - \frac{5}{2}}{\frac{1}{16} - \frac{5}{2} + 1}
     = \frac{-\frac{1}{2}}{\frac{1}{16} - \frac{10}{4} + 1}
     = \frac{-\frac{1}{2}}{-\frac{27}{16}}
     = \frac{8}{27}
     \]
   
2. **Combine the two expressions**:
   \[
   \lim_{{x \to \frac{1}{2}}} \left( \frac{3}{40} - \frac{8}{27} \right)
   \]

   To subtract these two fractions, we first find a common denominator:
   - The least common denominator (LCD) of 40 and 27 is 1080.
   - Convert each fraction:
     \[
     \frac{3}{40} = \frac{81}{1080}, \quad \frac{8}{27} = \frac{320}{1080}
     \]
   
3. **Subtract the fractions**:
   \[
   \frac{81}{1080} - \frac{320}{1080} = \frac{-239}{1080}
   \]

Thus, the limit is:

\[
\boxed{\frac{-239}{1080}}
\]

Would you like further clarifications on this process?

Here are five related questions to enhance your understanding:
1. How do you approach finding limits involving complex fractions?
2. What is the importance of substituting directly in a limit problem?
3. How do you find a least common denominator when subtracting fractions?
4. When is it appropriate to factor in order to simplify a limit problem?
5. How do you handle indeterminate forms that involve subtraction?

**Tip**: Always simplify

What is the limit of the expression lim_{x → 1/2} [(2x^3 - 1) / ((8x^2 + 2)(5x - 5)) - (2 - 5x) / (x^4 - 5x + 1)]?

Discover how to calculate the limit of a complex fraction involving direct substitution at x = 1/2. This problem delves into limits, fraction operations, and simplification techniques. Perfect for students in Grades 11-12.

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