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

• (a)

• (b)

• (c)

• (d)

• (e)

• (f)

• (g)

• (h)

Background

Topic: Limits from Graphs

This question tests your ability to interpret and evaluate limits using the graph of a function. You will need to analyze the behavior of the function as x approaches specific values, including from the left and right, and as x approaches infinity or negative infinity.

Key Terms and Concepts:

• Limit: The value a function approaches as the input (x) approaches a certain value.

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

• Limit at infinity: or .

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

Step-by-Step Guidance

1. For each limit, identify the x-value or direction (left, right, infinity) you are approaching. Look at the graph to see how the function behaves as x gets close to that value.

2. For limits at infinity ( or ), 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 (e.g., or ), trace the graph from the left and right sides of the specified x-value. Check for jumps, holes, or vertical asymptotes.

4. For , compare the left- and right-hand limits. If they are equal, the limit exists; if not, the limit does not exist at that point.

5. For limits at specific points (e.g., , , ), check if the function is continuous at that point or if there is a discontinuity. If there is a hole or jump, use the graph to determine the value the function approaches from both sides.

Try solving on your own before revealing the answer!