Consider the following statements:
I. `underset(xrarr0)lim (x^(2))/(x)` exists.
II. `((x^(2))/(x))` is not continuous at x = 0.
III. `underset(xrarr0)(lim)(|x|)/(x)` does not exist
Which of the above statements are correct?

Options

A. A) I, II, and III
B. B) I and II
C. C) II and III
D. D) I and III

Answer

To determine which of the statements are correct, we will analyze each statement step by step.

### Step 1: Analyze Statement I
**Statement I:** \( \lim_{x \to 0} \frac{x^2}{x} \) exists.

1. Simplify the expression:
\[ \frac{x^2}{x} = x \quad \text{(for } x \neq 0\text{)} \]

2. Now, find the limit as \( x \) approaches 0:
\[ \lim_{x \to 0} x = 0 \]

3. Since the limit exists and equals 0, **Statement I is true**.

### Step 2: Analyze Statement II
**Statement II:** \( \frac{x^2}{x} \) is not continuous at \( x = 0 \).

1. The function \( \frac{x^2}{x} \) simplifies to \( x \) for \( x \neq 0 \).

2. The function \( x \) is a polynomial function, which is continuous everywhere, including at \( x = 0 \).

3. Therefore, **Statement II is false**.

### Step 3: Analyze Statement III
**Statement III:** \( \lim_{x \to 0} \frac{|x|}{x} \) does not exist.

1. We need to evaluate the limit from both sides:
- For \( x > 0 \):
\[ \frac{|x|}{x} = \frac{x}{x} = 1 \]
- For \( x < 0 \):
\[ \frac{|x|}{x} = \frac{-x}{x} = -1 \]

2. Now, calculate the left-hand limit and right-hand limit:
- Left-hand limit:
\[ \lim_{x \to 0^-} \frac{|x|}{x} = -1 \]
- Right-hand limit:
\[ \lim_{x \to 0^+} \frac{|x|}{x} = 1 \]

3. Since the left-hand limit and right-hand limit are not equal:
\[ -1 \neq 1 \]
Therefore, the limit does not exist.

4. Hence, **Statement III is true**.

### Conclusion
- **Statement I**: True
- **Statement II**: False
- **Statement III**: True

The correct statements are I and III.

### Final Answer
The correct statements are **I and III**.

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