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We are given the limit expression:

\[
\lim_{{x \to 3}} \frac{2x - 6}{7x - 21}
\]

### Step-by-Step Solution:

1. **Evaluate the expression by substituting \( x = 3 \)**:

   \[
   \text{Numerator: } 2(3) - 6 = 6 - 6 = 0
   \]
   \[
   \text{Denominator: } 7(3) - 21 = 21 - 21 = 0
   \]

   Since substituting \( x = 3 \) yields a \( \frac{0}{0} \) form, we must simplify the expression further.

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

   \[
   \text{Numerator: } 2x - 6 = 2(x - 3)
   \]
   \[
   \text{Denominator: } 7x - 21 = 7(x - 3)
   \]

   Thus, the limit expression becomes:

   \[
   \lim_{{x \to 3}} \frac{2(x - 3)}{7(x - 3)}
   \]

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

   \[
   \lim_{{x \to 3}} \frac{2}{7} = \frac{2}{7}
   \]

### Final Answer:
\[
\lim_{{x \to 3}} \frac{2x - 6}{7x - 21} = \frac{2}{7}
\]

Would you like a more detailed explanation or have any other questions?

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Here are some related questions to deepen understanding:

1. What happens if we directly substitute \( x = 3 \) into the unsimplified expression?
2. How do you identify whether a limit results in an indeterminate form?
3. Can all indeterminate forms be solved by factoring?
4. What alternative methods can be used to resolve \( \frac{0}{0} \) forms?
5. How do we handle cases when both numerator and denominator cannot be factored directly?

**Tip:** Always check if substituting directly into a limit yields a well-defined number. If it results in \( \frac{0}{0} \), explore simplification options such as factoring or rationalizing.

Evaluate \( \lim_{{x \to 3}} \frac{2x - 6}{7x - 21} \) and determine the value.

Learn how to evaluate the limit as x approaches 3 for the expression (2x - 6)/(7x - 21). This problem involves factoring and simplifying indeterminate forms to find the limit. Suitable for high school students in Grades 10-12.

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