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The image contains a problem asking to evaluate the limit of functions as \( x \to 0 \). Here's a breakdown of the tasks:

### Problem Statement
- **Instruction**: "Estimate the limit of \( f(x) \) as \( x \to 0 \) for the following functions (if they exist). Use a calculator if necessary."
- **Notation**: \( |a| \) represents the greatest integer less than or equal to \( a \).

### Functions to analyze:
1. **(a)** \( f(x) = x \left\lfloor \frac{1}{x} \right\rfloor \), where \( \left\lfloor a \right\rfloor \) represents the greatest integer less than or equal to \( a \).
   - Example provided: \( \left\lfloor 2.7 \right\rfloor = 2 \).

2. **(b)** \( f(x) = x \left\lceil \frac{1}{x} \right\rceil \), where \( \left\lceil a \right\rceil \) represents the smallest integer greater than or equal to \( a \).
   - Example provided: \( \left\lceil 2.3 \right\rceil = 2 \).

3. **(c)** \( f(x) = |x| \), where \( |x| \) represents the absolute value of \( x \).

### Solutions:

#### (a) \( f(x) = x \left\lfloor \frac{1}{x} \right\rfloor \)

For this function, as \( x \to 0 \) from the positive side, \( \frac{1}{x} \to \infty \), and \( \left\lfloor \frac{1}{x} \right\rfloor \) will be a large integer. So,
\[
f(x) = x \left\lfloor \frac{1}{x} \right\rfloor.
\]
For small positive \( x \), the value \( \left\lfloor \frac{1}{x} \right\rfloor \) will be close to \( \frac{1}{x} \), making \( f(x) \) approximately 1.

When \( x \to 0 \) from the negative side, \( \frac{1}{x} \to -\infty \), and \( \left\lfloor \frac{1}{x} \right\rfloor \) will be a large negative integer. So, \( f(x) \) will be close to -1.

Thus, the limit does not exist, as the behavior differs from the left and the right.

#### (b) \( f(x) = x \left\lceil \frac{1}{x} \right\rceil \)

For small positive \( x \), \( \left\lceil \frac{1}{x} \right\rceil \approx \frac{1}{x} \), so \( f(x) \approx 1 \).

For small negative \( x \), \( \left\lceil \frac{1}{x} \right\rceil \approx \frac{1}{x} \), which gives \( f(x) \approx -1 \).

Again, the limit does not exist as \( x \to 0 \), since the values from the left and the right differ.

#### (c) \( f(x) = |x| \)

The function \( f(x) = |x| \) is continuous and well-behaved. As \( x \to 0 \), the value of \( |x| \to 0 \).

Thus, the limit is:
\[
\lim_{x \to 0} |x| = 0.
\]

### Summary:
- For (a), the limit does not exist.
- For (b), the limit does not exist.
- For (c), the limit is 0.

Would you like further clarification on any of these steps or explanations?

### Follow-up questions:
1. Why does the floor function behave differently for positive and negative values of \( x \)?
2. How does the ceiling function affect the limit as \( x \to 0 \)?
3. Can you think of a modification to \( f(x) \) in parts (a) or (b) to make the limit exist?
4. What is the significance of the absolute value in part (c)?
5. Can the limit of \( f(x) = x \cdot g(x) \) exist if \( g(x) \) oscillates rapidly?

### Tip:
Always check the behavior of the function from both the left and right sides when computing limits at a point of discontinuity.

Taksirlah nilai lim x→0 f(x) dari fungsi-fungsi berikut: a) f(x) = x⌊1/x⌋, b) f(x) = x⌈1/x⌉, c) f(x) = |x|.

This problem focuses on evaluating the limit of functions as x approaches 0, involving the floor function (⌊ ⌋), ceiling function (⌈ ⌉), and absolute value. Key mathematical concepts include limit evaluation and properties of these special functions. Suitable for undergraduate math students.

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