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The problem asks for the limit of the expression:

\[
\lim_{{x \to 6}} \frac{{3x - 18}}{{2x - 12}}
\]

### Step-by-Step Solution:

1. **Substitute** \( x = 6 \) into the expression directly:

   \[
   \frac{{3(6) - 18}}{{2(6) - 12}} = \frac{{18 - 18}}{{12 - 12}} = \frac{0}{0}
   \]

   This results in an indeterminate form \( \frac{0}{0} \), so we need to simplify the expression further.

2. **Factor the numerator and denominator**:

   - The numerator \( 3x - 18 \) can be factored as \( 3(x - 6) \).
   - The denominator \( 2x - 12 \) can be factored as \( 2(x - 6) \).

   So, the expression becomes:

   \[
   \frac{{3(x - 6)}}{{2(x - 6)}}
   \]

3. **Cancel the common factor** \( (x - 6) \):

   \[
   \frac{3}{2}
   \]

4. **Evaluate the limit**:

   Since the remaining expression is constant, the limit as \( x \to 6 \) is simply:

   \[
   \lim_{{x \to 6}} \frac{3}{2} = \frac{3}{2}
   \]

### Final Answer:
\[
\lim_{{x \to 6}} \frac{{3x - 18}}{{2x - 12}} = \frac{3}{2}
\]

Would you like further details on any of the steps? Here are some related questions for practice:

1. What is an indeterminate form in limits, and why does \( \frac{0}{0} \) occur?
2. How do you simplify rational expressions to evaluate limits?
3. What other methods can be used to solve limits that result in \( \frac{0}{0} \)?
4. Can you always cancel terms in limits, or are there exceptions?
5. What role does factoring play in solving limits?

**Tip**: Always try direct substitution first when solving limits, as it helps to identify if further simplification is needed!

Determine the following limit: lim (x → 6) [(3x - 18) / (2x - 12)]

Learn how to solve the limit of (3x - 18) / (2x - 12) as x approaches 6 using factorization and cancellation techniques. This problem covers fundamental calculus concepts suitable for students in Grades 11-12 or Calculus 1.

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