Consider the function: f(x)=|x|x: (i) limx→0+f(x)=1 (ii) limx→0−f(x)=−1 (iii)limx→0f(x) does not exist
 A. None is true
 B. (i) and (iii) are true
 C. Both (i) and (ii) are false
 D. All (i), (ii) and (iii) are true

Let's analyze the given function \( f(x) = \frac{|x|}{x} \) and determine the limits.

### Step 1: Analyze the function for \( x > 0 \) (Right-hand limit)
When \( x > 0 \), \( |x| = x \). So the function becomes:
\[
f(x) = \frac{x}{x} = 1
\]
Therefore:
\[
\lim_{x \to 0^+} f(x) = 1
\]
This matches statement (i).

### Step 2: Analyze the function for \( x < 0 \) (Left-hand limit)
When \( x < 0 \), \( |x| = -x \). So the function becomes:
\[
f(x) = \frac{-x}{x} = -1
\]
Therefore:
\[
\lim_{x \to 0^-} f(x) = -1
\]
This matches statement (ii).

### Step 3: Determine the two-sided limit as \( x \to 0 \)
For the two-sided limit \( \lim_{x \to 0} f(x) \), the left-hand limit and the right-hand limit are different:
\[
\lim_{x \to 0^+} f(x) = 1 \quad \text{and} \quad \lim_{x \to 0^-} f(x) = -1
\]
Since the left-hand limit and right-hand limit are not equal, the limit does not exist.

This matches statement (iii).

### Conclusion
The correct answer is:

**D. All (i), (ii), and (iii) are true**

Would you like further details on this topic, or do you have any other questions?

Here are some related questions:
1. How do you evaluate limits involving absolute value functions?
2. What is the significance of one-sided limits in determining the existence of a limit?
3. How does the behavior of a function near a point affect the limit at that point?
4. Can you provide an example where the two-sided limit exists but differs from one-sided limits?
5. What does it mean for a limit to "not exist"?
6. How do discontinuities impact the existence of limits?
7. What is the difference between a jump discontinuity and a removable discontinuity?
8. How do piecewise functions relate to the concept of limits?

**Tip:** When evaluating limits, always consider the behavior of the function from both the left and the right sides of the point of interest.

Explore the analysis of limits involving absolute value functions using the example of f(x) = |x| / x. Learn about the behavior of one-sided limits and why the two-sided limit may not exist. Suitable for advanced high school students and those preparing for calculus.

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