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The given mathematical exercise consists of the function \( f(x) = \frac{x^2 - 2x}{x} \). Let's solve the three parts step by step:

### a) Finding the domain \( A \) of the function

The function is defined as:

\[
f(x) = \frac{x^2 - 2x}{x}
\]

We know that the function is not defined when the denominator is zero. So, we set \( x \neq 0 \) to ensure the function is defined. The domain of the function is therefore all real numbers except zero:

\[
A = \mathbb{R} \setminus \{0\}
\]

### b) Prove that for every \( x \in A \), \( f(x) = x - 2 \)

We simplify the function \( f(x) \):

\[
f(x) = \frac{x^2 - 2x}{x} = \frac{x(x - 2)}{x}
\]

For \( x \neq 0 \), we can cancel the \( x \) terms:

\[
f(x) = x - 2
\]

Thus, we have proven that \( f(x) = x - 2 \) for all \( x \in A \).

### c) Calculate \( \lim_{x \to 0} f(x) \)

From part (b), we know that \( f(x) = x - 2 \) for \( x \in A \). Therefore, we calculate the limit as \( x \to 0 \):

\[
\lim_{x \to 0} f(x) = \lim_{x \to 0} (x - 2)
\]

This is a simple linear function, and as \( x \to 0 \), the limit becomes:

\[
\lim_{x \to 0} (x - 2) = -2
\]

Thus, \( \lim_{x \to 0} f(x) = -2 \).

---

Would you like more details on any of the parts or further clarifications?

Here are 5 related questions for practice:
1. What happens to the function if you modify it to \( f(x) = \frac{x^2 - 3x}{x} \)?
2. How would you approach finding the derivative of \( f(x) = x - 2 \)?
3. What would the limit be if the function were \( f(x) = x^2 - 2x \) as \( x \to 0 \)?
4. What is the range of \( f(x) = x - 2 \)?
5. How would the domain change if \( f(x) = \frac{x^2 - 2x}{x^2} \)?

**Tip:** When simplifying rational expressions, always check for domain restrictions before canceling terms.

Given the function f(x) = (x^2 - 2x) / x. 
a) Find the domain A of the function. 
b) Prove that for all x ∈ A: f(x) = x - 2. 
c) Compute the limit lim (x → 0) f(x).

Learn how to find the domain of the function f(x) = (x^2 - 2x) / x, prove its simplified form, and compute the limit as x approaches 0. This problem covers key algebraic concepts and limit theorems. Suitable for students in Grades 9-11.

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