To solve the expression $\frac{f(a+h) - f(a)}{h}$ for the function $f(x) = \frac{x - 4}{x - 5}$, let's go through the process step-by-step:

1. Calculate $f(a+h)$:

Substitute $a+h$ into $f(x)$: $f(a+h) = \frac{(a+h) - 4}{(a+h) - 5}$ $f(a+h) = \frac{a + h - 4}{a + h - 5}$

2. Calculate $f(a)$:

Substitute $a$ into $f(x)$: $f(a) = \frac{a - 4}{a - 5}$

3. Set up the difference quotient $\frac{f(a+h) - f(a)}{h}$:

$\frac{f(a+h) - f(a)}{h} = \frac{\frac{a + h - 4}{a + h - 5} - \frac{a - 4}{a - 5}}{h}$

4. Combine the two fractions in the numerator:

To combine the two fractions, we need a common denominator. The common denominator will be $(a+h - 5)(a - 5)$. Let's rewrite the expression:

$f(a+h) - f(a) = \frac{(a+h - 4)(a - 5) - (a - 4)(a+h - 5)}{(a+h - 5)(a - 5)}$

Now, expand both numerators:

$(a+h - 4)(a - 5) = (a+h)(a-5) - 4(a-5) = a^2 - 5a + ha - 5h - 4a + 20$

$(a - 4)(a+h - 5) = (a-4)(a+h-5) = a^2 - 5a + ah - 5h - 4a + 20$

Now subtract the two numerators: $\left(a^2 - 5a + ha - 5h - 4a + 20 \right) - \left(a^2 - 5a + ha - 5h - 4a + 20 \right) = 0$

5. Simplify the numerator and the quotient:

The result simplifies to zero, which implies that: $\frac{f(a+h) - f(a)}{h} = 0$

Thus, the final answer is:

$\boxed{0}$

Would you like more details on any step, or clarification on how this result came about?

Here are 5 related questions to deepen your understanding:

1. What is the geometric meaning of the difference quotient in calculus?
2. How does the difference quotient relate to the concept of the derivative?
3. Can you compute the derivative of $f(x) = \frac{x-4}{x-5}$ using first principles?
4. What are the points at which the function $f(x)$ is not defined?
5. How does the behavior of $f(x)$ change as $x$ approaches 5?

Tip: When calculating a difference quotient, always ensure that the denominators in the fractional expressions are fully simplified before proceeding to subtraction.