https://file.aiquickdraw.com/mathbot/file/17365935484845ub50782.jpg

Let me analyze the solution in the uploaded image and verify the calculations step by step.

### Given Problem:
\[
\lim_{x \to \frac{1}{2}} \frac{8x^3 - 1}{6x^2 - 5x + 1}
\]

From the solution shown:
1. Initially, substituting \(x = \frac{1}{2}\) into the expression results in an indeterminate form \(\frac{0}{0}\).
2. To resolve this, polynomial division and simplification have been used.

---

### Step-by-Step Verification:

#### Step 1: Substitute \(x = \frac{1}{2}\)
We calculate numerator and denominator separately:
- Numerator: \(8x^3 - 1\)
  \[
  8\left(\frac{1}{2}\right)^3 - 1 = 8 \cdot \frac{1}{8} - 1 = 1 - 1 = 0
  \]
- Denominator: \(6x^2 - 5x + 1\)
  \[
  6\left(\frac{1}{2}\right)^2 - 5\left(\frac{1}{2}\right) + 1 = 6 \cdot \frac{1}{4} - \frac{5}{2} + 1 = \frac{3}{2} - \frac{5}{2} + 1 = 0
  \]
Thus, substituting \(x = \frac{1}{2}\) gives \(\frac{0}{0}\), an indeterminate form.

---

#### Step 2: Simplify Using Polynomial Division
Perform the polynomial division of:
\[
\frac{8x^3 - 1}{6x^2 - 5x + 1}
\]

The solution shows the division step-by-step:

1. **Divide the leading terms**:
   - Divide \(8x^3\) by \(6x^2\): \(\frac{8x}{6} = \frac{4x}{3}\).
   - Multiply \(\frac{4x}{3}(6x^2 - 5x + 1)\), and subtract from the numerator.

2. **Result of division**:
   After dividing, the simplified result appears to be \((4x^2 - 2x + 1) / (3x - 1)\).

---

#### Step 3: Simplify the Limit Expression
The solution then simplifies:
\[
\lim_{x \to \frac{1}{2}} \frac{4x^2 - 2x + 1}{3x - 1}
\]

Substitute \(x = \frac{1}{2}\) into the simplified expression:
- Numerator: \(4x^2 - 2x + 1\)
  \[
  4\left(\frac{1}{2}\right)^2 - 2\left(\frac{1}{2}\right) + 1 = 4 \cdot \frac{1}{4} - 1 + 1 = 1
  \]
- Denominator: \(3x - 1\)
  \[
  3\left(\frac{1}{2}\right) - 1 = \frac{3}{2} - 1 = \frac{1}{2}
  \]

Thus, the limit is:
\[
\frac{1}{\frac{1}{2}} = 2
\]

---

### Final Answer:
The limit is correctly calculated as:
\[
\boxed{2}
\]

---

### Do you have any questions? Would you like a detailed explanation of any specific step?  

Here are **5 related questions** to expand your understanding:  
1. How is polynomial division used to simplify rational expressions?  
2. Why does substituting directly into the limit sometimes yield indeterminate forms?  
3. What are alternative methods for solving limits aside from polynomial division?  
4. How does the degree of the numerator and denominator affect the behavior of limits?  
5. What are the common cases where L’Hôpital’s Rule is useful in limits?  

**Tip**: Always simplify the numerator and denominator fully before substituting into a limit to avoid unnecessary complications!

lim (x -> 1/2) (8x^3 - 1) / (6x^2 - 5x + 1)

Learn how to calculate the limit of a rational function as x approaches 1/2. This problem demonstrates polynomial division, simplification, and applying the limit theorem for indeterminate forms. Suitable for students in Grade 11-12.

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