https://file.aiquickdraw.com/mathbot/file/1725942992870tqhf1hqx.jpg

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.

Use the limit definition to find the derivative of the function f(x) = x - 2. First, find (f(x + h) - f(x)) / h. Next, simplify the numerator. Divide out the h. So now, find the limit as h approaches 0.

Learn how to find the derivative of the linear function f(x) = x - 2 using the limit definition. This step-by-step solution covers key calculus concepts, including simplifying expressions and applying limits. Suitable for high school students in Grades 11-12.

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