Let's solve the problem step by step using the limit definition of a derivative.

Given function:

$f(x) = x - 2$

Step 1: Apply the difference quotient formula:

$\frac{f(x+h) - f(x)}{h}$

First, we need to calculate $f(x + h)$: $f(x+h) = (x+h) - 2 = x + h - 2$

Now, substitute $f(x + h)$ and $f(x)$ into the difference quotient: $\frac{f(x+h) - f(x)}{h} = \frac{(x + h - 2) - (x - 2)}{h}$

Step 2: Simplify the numerator:

$(x + h - 2) - (x - 2) = x + h - 2 - x + 2 = h$

So the expression becomes: $\frac{h}{h}$

Step 3: Divide out the $h$:

$\frac{h}{h} = 1 \quad \text{(for } h \neq 0\text{)}$

Step 4: Find the limit as $h \to 0$:

$\lim_{h \to 0} 1 = 1$

Conclusion:

The derivative of $f(x) = x - 2$ is: $f'(x) = 1$

Would you like further details on any part of the process?

Here are five related questions for further exploration:

1. How is the limit definition of the derivative applied in general?
2. What would change if the function were more complex, such as $f(x) = x^2 - 2$?
3. Why is the limit necessary in defining the derivative?
4. Can this approach be extended to higher-order derivatives?
5. How does this derivative help in understanding the function's graph?

Tip: The limit definition of a derivative provides the slope of the tangent line at any point on the function, which is key to understanding rates of change.