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}{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}$$
   • In the second term: = \frac{2 - \frac{5}{2}}{\frac{1}{16} - \frac{5}{2} + 1} = \frac{-\frac{1}{2}}{\frac{1}{16} - \frac{10}{4} + 1} = \frac{-\frac{1}{2}}{-\frac{27}{16}} = \frac{8}{27}$$

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