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?

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.