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To solve the given problem, we need to evaluate the difference quotient for the function \( f(x) = \frac{5}{x - 3} \) at \( x = 6 \), and then take the limit as \( h \to 0 \). Let's go through the steps:

1. **Find \( f(6+h) \) and \( f(6) \):**
   - \( f(6+h) = \frac{5}{(6+h) - 3} = \frac{5}{h+3} \)
   - \( f(6) = \frac{5}{6-3} = \frac{5}{3} \)

2. **Form the difference quotient:**
   \[
   \frac{f(6+h) - f(6)}{h} = \frac{\frac{5}{h+3} - \frac{5}{3}}{h}
   \]

3. **Simplify the expression:**
   To combine the fractions in the numerator:
   \[
   \frac{5}{h+3} - \frac{5}{3} = \frac{5 \cdot 3 - 5 \cdot (h+3)}{3(h+3)} = \frac{15 - 5h - 15}{3(h+3)} = \frac{-5h}{3(h+3)}
   \]

   Thus, the difference quotient becomes:
   \[
   \frac{-5h}{3(h+3)} \cdot \frac{1}{h} = \frac{-5}{3(h+3)}
   \]

4. **Evaluate the limit as \( h \to 0 \):**
   \[
   \lim_{h \to 0} \frac{-5}{3(h+3)} = \frac{-5}{3 \cdot 3} = \frac{-5}{9}
   \]

Therefore, the final answer is \( \frac{-5}{9} \).

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

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Here are five related questions for further exploration:
1. How do you interpret the meaning of a derivative at a point in terms of a function’s graph?
2. What does the negative sign of the derivative in this example signify about the function's behavior at \( x = 6 \)?
3. How would you approach finding the derivative of a more complicated rational function using the difference quotient?
4. Can this limit be evaluated using L'Hôpital's Rule, and if so, how?
5. What role does continuity play in evaluating the limit of a difference quotient?

**Tip:** When simplifying a difference quotient, always look for common denominators to combine fractions effectively.

Evaluate the limit of the difference quotient for f(x) = 5/(x - 3) at x = 6 and take the limit as h approaches 0.

Learn how to evaluate the limit of the difference quotient for the rational function f(x) = 5/(x - 3) at x = 6. This step-by-step guide simplifies the difference quotient and calculates the limit, showcasing key concepts of limits and rational functions. Suitable for students in Grades 10-12.

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