The given problem asks for the limit of the following expression as $x$ approaches 8:

$\lim_{{x \to 8}} \frac{{x - 8}}{{8(x - 8)}}$

Step-by-Step Solution:

1. Simplify the expression: The numerator and denominator both contain the term $(x - 8)$, which we can cancel out, provided that $x \neq 8$ to avoid division by zero.

   $\frac{{x - 8}}{{8(x - 8)}} = \frac{1}{8}, \quad \text{for} \, x \neq 8.$

2. Evaluate the limit: Since the expression simplifies to $\frac{1}{8}$, and this holds for all $x \neq 8$, the limit as $x \to 8$ is simply:

   $\lim_{{x \to 8}} \frac{1}{8} = \frac{1}{8}.$

Final Answer:

The limit is $\frac{1}{8}$.

Would you like more details or have any questions? Here are 5 related questions that can expand on this concept:

1. What happens if the expression cannot be simplified (e.g., if cancellation isn't possible)?
2. How do limits behave when approaching values where the denominator goes to zero?
3. Can you explain what it means for a limit to "not exist"?
4. How does L'Hopital's Rule help in cases like these where direct substitution leads to indeterminate forms?
5. Can limits of more complicated functions also be simplified in this way?

Tip: Always check for indeterminate forms like $\frac{0}{0}$ before canceling terms in a limit problem.