Q2. The graph of is shown above. Find the following limits:

Background

Topic: Limits from a Graph

This question tests your ability to evaluate limits by interpreting the behavior of a function as approaches a particular value, using a graph. You will need to consider both one-sided and two-sided limits, as well as limits at infinity.

Key Terms and Concepts:

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

• One-sided limits: (from the left), (from the right).

• Limit at infinity: or describes the end behavior of the function.

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

Step-by-Step Guidance

1. For each limit, carefully observe the graph as approaches the specified value. Pay attention to whether the function approaches a finite value, infinity, or does not settle to a single value.

2. For one-sided limits (like or ), look at the behavior of the graph as approaches from only one side (left or right).

3. For two-sided limits (like ), check if the left-hand and right-hand limits are equal. If they are not, the two-sided limit does not exist.

4. For limits at infinity, observe the end behavior of the graph as becomes very large (positive or negative). Does the function approach a horizontal asymptote, or does it increase/decrease without bound?

5. If the graph has a jump, hole, or vertical asymptote at the point, consider whether the limit exists or not. If the function approaches different values from the left and right, the limit does not exist at that point.

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