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The problem in the image deals with finding the limit of the expression \(\lim_{x \to 2} \frac{(3x - 1)^3 - (x + 3)^3}{x^3 - 8}\).

### Solution breakdown:

1. **Direct substitution**:
   - First, the solution substitutes \(x = 2\) into the expression, leading to the form \(\frac{0}{0}\), which is indeterminate. This indicates that a different approach is needed to evaluate the limit.

2. **Factoring the numerator**:
   - The solution proceeds by factoring the cubic expressions in the numerator. 
   - Both \((3x - 1)^3\) and \((x + 3)^3\) are expanded and simplified.
   - After factoring out common terms, the numerator can be expressed in a factored form that cancels out the indeterminate part with the denominator.

3. **Simplifying**:
   - The factor \((x - 2)\) is factored out of the numerator and cancels with the corresponding factor in the denominator, leaving a simpler expression to substitute \(x = 2\).

4. **Final substitution**:
   - After canceling and simplifying, the limit can be evaluated by direct substitution, which yields the final result \( \frac{25}{2} \).

Thus, the final answer is:

\[
\lim_{x \to 2} \frac{(3x - 1)^3 - (x + 3)^3}{x^3 - 8} = \frac{25}{2}
\]

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Would you like more details on any step? 

Here are 5 questions to expand your understanding:

1. Why is it necessary to factor the expression when the direct substitution gives \( \frac{0}{0} \)?
2. How is the difference of cubes formula used in this problem?
3. What are the steps to expand \((3x - 1)^3\) and \((x + 3)^3\)?
4. How do we factor out the common terms from the numerator and denominator?
5. What would happen if we used L'Hopital's Rule instead of factoring?

**Tip**: When facing indeterminate forms like \(\frac{0}{0}\), look for factoring or simplifying techniques to resolve the limit.

Find the limit lim_{x -> 2} [(3x - 1)^3 - (x + 3)^3] / (x^3 - 8).

This problem focuses on solving a limit involving algebraic expressions. Learn how to factor and simplify the expression to find the limit as x approaches 2 for [(3x - 1)^3 - (x + 3)^3] / (x^3 - 8). Suitable for Grades 11-12.

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