Q2. The graph of y = f(x) is shown above. Find the following limits:

Background

Topic: Limits from a Graph

This question tests your ability to interpret a graph and determine the value of limits at various points, including at infinity, at points of discontinuity, and at specific x-values. You will need to understand one-sided limits, jump discontinuities, and infinite limits.

Key Terms:

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

• One-sided limit: The value the function approaches from one side (left or right) of a point.

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

• Limit at infinity: The value the function approaches as x goes to positive or negative infinity.

Step-by-Step Guidance

1. Examine the graph carefully for each x-value or direction (e.g., , , , , etc.). Identify whether the function is continuous, has a jump, or goes to infinity at each point.

2. For limits at infinity ( and ), observe the end behavior of the graph. Does the function approach a horizontal asymptote, or does it increase/decrease without bound?

3. For one-sided limits at , look at the values the function approaches from the left () and from the right (). If the values are different, the two-sided limit does not exist.

4. For limits at points of discontinuity (such as or ), check if the function jumps, has a hole, or goes to infinity. Use the graph to estimate the value or determine if the limit does not exist.

5. For other specific x-values (like , ), trace the graph to see what value the function approaches as x gets close to those points.

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