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Let's analyze the graph of \( y = f(x) \) and answer each of the provided questions:

### (a) Does \( \lim_{x \to 0} f(x) \) exist?

From the graph, observe the behavior of \( f(x) \) as \( x \) approaches 0 from both sides (left and right):

- As \( x \) approaches 0 from the left (\( x \to 0^- \)), the function decreases without bound, tending towards negative infinity.
- As \( x \) approaches 0 from the right (\( x \to 0^+ \)), the function increases without bound, tending towards positive infinity.

Since the left-hand and right-hand limits are not equal and both tend towards infinity in opposite directions, the limit does not exist.

**Answer: No**

### (b) Is \( f(x) \) continuous at \( x = 0 \)?

For a function to be continuous at a point, three conditions must be satisfied:
1. The function must be defined at that point.
2. The limit as \( x \) approaches the point must exist.
3. The value of the function at the point must equal the limit as \( x \) approaches the point.

From the graph, we see that:
- The limit as \( x \to 0 \) does not exist.
- Additionally, it seems that \( f(0) \) is undefined, as there is no specific point shown for \( x = 0 \).

Therefore, \( f(x) \) is not continuous at \( x = 0 \).

**Answer: No**

### (c) Is \( f(x) \) differentiable at \( x = 0 \)?

A function is differentiable at a point if it is continuous and has a well-defined slope (tangent) at that point. Since we already determined that \( f(x) \) is not continuous at \( x = 0 \), it cannot be differentiable there.

**Answer: No**

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Let me know if you'd like more details or have any questions! Here are some related questions to explore further:

1. What are the necessary conditions for differentiability at a point?
2. Can a function be continuous but not differentiable at a point? If so, give an example.
3. What is the relationship between left-hand and right-hand limits for continuity?
4. How does the existence of vertical asymptotes affect the limit at a point?
5. What are common examples of functions with discontinuities at a point?

**Tip**: Always check the behavior of the function from both sides when evaluating limits!

Analyze the graph of y = f(x) to determine: (a) if \( \lim_{x \to 0} f(x) \) exists, (b) if f(x) is continuous at x = 0, and (c) if f(x) is differentiable at x = 0.

This problem explores the graph of a function f(x) and evaluates whether the limit at x = 0 exists, if the function is continuous at x = 0, and whether it is differentiable at x = 0. It covers key concepts in calculus, including limits, continuity, and differentiability.

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