Q1. Analyze the graph of the function f to answer the following questions:

Background

Topic: Limits, Continuity, and Differentiability

This question tests your understanding of how to interpret a graph to determine limits, points of continuity/discontinuity, and differentiability. You will use visual cues from the graph to answer questions about the behavior of the function at specific points.

Key Terms and Concepts:

• Limit: The value that a function approaches as the input approaches a certain point.

• Continuity: A function is continuous at a point if the limit exists and equals the function's value at that point.

• Discontinuity: A point where the function is not continuous (can be jump, removable, or infinite).

• Differentiability: A function is differentiable at a point if it is smooth (no sharp corners or cusps) and continuous there.

• Vertical Asymptote: A line where the function grows without bound as it approaches a certain x-value.

Step-by-Step Guidance

1. For each limit question (e.g., ), examine the graph near the specified x-value. Look at the left and right sides to see if the function approaches a specific value or diverges.

2. If the function jumps, has a hole, or diverges at the point, consider whether the limit exists or if you should use DNE (Does Not Exist).

3. For continuity questions, check if the function is unbroken at the specified x-value. The function must be defined, the limit must exist, and both must be equal.

4. For differentiability, look for sharp corners, cusps, or discontinuities. If the function is not smooth or not continuous at the point, it is not differentiable there.

5. For vertical asymptotes, identify where the function grows very large (positive or negative) as x approaches a certain value.

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