We are tasked with finding the limit:

$\lim_{{x \to 17}} \frac{\sqrt{x - 8} - 2}{x}.$

Step 1: Direct Substitution

The question asks to apply direct substitution. First, we substitute $x = 17$ directly into the expression:

$\frac{\sqrt{17 - 8} - 2}{17}.$

Simplify inside the square root:

$\frac{\sqrt{9} - 2}{17} = \frac{3 - 2}{17} = \frac{1}{17}.$

Thus, the limit is:

$\boxed{\frac{1}{17}}.$

Would you like more details or have further questions? Here are five related questions for practice:

1. What happens if the limit approaches a point where the denominator equals zero?
2. How do you handle indeterminate forms in limits?
3. What are the different techniques to solve limits other than direct substitution?
4. Can you apply L'Hopital's Rule to this limit if direct substitution doesn't work?
5. How do limits relate to continuity at a point?

Tip: Always check if direct substitution results in a defined value before using more advanced methods like factoring or L'Hopital's Rule.