To evaluate the limit:

$\lim_{{x \to 9}} \frac{9 - x}{x - 3}$

Substitute $x = 9$ directly into the expression:

$\frac{9 - 9}{9 - 3} = \frac{0}{6} = 0$

Thus, the limit is:

$\boxed{0}$

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

Here are some related questions:

1. What is a limit, and how do we calculate it for more complex functions?
2. How does substitution work when evaluating limits?
3. What happens if direct substitution results in an indeterminate form (e.g., $\frac{0}{0}$)?
4. What are one-sided limits, and how are they useful?
5. How does continuity of a function affect the calculation of limits?

Tip: Always try direct substitution first when evaluating a limit. If it leads to an indeterminate form, then consider using algebraic simplification, factoring, or L'Hôpital's rule.