Let's analyze and solve the given mathematical problem from your image.

We have the function $f(x) = \frac{x^2 - x}{x}$.

Part (a): Domain of $f(x)$

To find the domain, we need to identify any values of $x$ that would make the function undefined. The function is a rational expression, and it is undefined when the denominator is 0.

Since the denominator is $x$, the function is undefined at $x = 0$.

Domain of $f(x)$: $A = \mathbb{R} \setminus \{0\}$ This means the domain is all real numbers except $x = 0$.

Part (b): Proof that $f(x) = x - 1$ for all $x \in A$

We simplify the function: $f(x) = \frac{x^2 - x}{x} = \frac{x(x - 1)}{x}.$ For $x \neq 0$, we can cancel out $x$ from the numerator and denominator: $f(x) = x - 1.$ Thus, for all $x \in A$, $f(x) = x - 1$, as required.

Part (c): Compute $\lim_{x \to 1} f(x)$

We need to find the limit of $f(x)$ as $x$ approaches 1. From part (b), we know that $f(x) = x - 1$, so: $\lim_{x \to 1} f(x) = \lim_{x \to 1} (x - 1).$ Substituting $x = 1$ into the expression: $\lim_{x \to 1} (x - 1) = 1 - 1 = 0.$ Thus, the limit is: $\lim_{x \to 1} f(x) = 0.$

Summary of Answers:

• (a): The domain is $A = \mathbb{R} \setminus \{0\}$.
• (b): $f(x) = x - 1$ for all $x \in A$.
• (c): $\lim_{x \to 1} f(x) = 0$.

Would you like more details on any of the steps or have any questions?

Here are 5 related questions to explore further:

1. How would the domain change if the denominator was $x + 2$ instead of $x$?
2. How can you verify the limit $\lim_{x \to 1} f(x)$ graphically?
3. What is the significance of removable discontinuities in rational functions like this one?
4. How do you handle limits when the function involves more complex expressions like trigonometric or exponential terms?
5. Can you compute the limit $\lim_{x \to 0} f(x)$ for the given function?

Tip: Always check for values of $x$ that make the denominator zero when finding the domain of a rational function.