Q5. For the function shown in the graph, answer each part for each of the following values of : :

• (a)

• (b)

• (c)

• (d)

• (e) Is in the domain of ? Why or why not?

• (f) Is continuous at ? Why or why not?

Background

Topic: Limits and Continuity from a Graph

This question tests your ability to interpret a graph to determine one-sided and two-sided limits, function values, domain, and continuity at specific points.

Key Terms and Formulas

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

• Two-sided limit: exists if and only if both one-sided limits exist and are equal

• Function value: is the value of the function at (if defined)

• Continuity at : is continuous at if is defined, exists, and

Step-by-Step Guidance

1. For each value of (), examine the graph of at and around .

2. To find , look at the -values as approaches from the left. For , look as approaches from the right.

3. Determine if the two one-sided limits are equal. If so, the two-sided limit exists and equals that value. If not, the two-sided limit does not exist.

4. Check if there is a filled (solid) dot at on the graph. If so, is defined and equals the -value of that dot. If there is an open circle, is not defined at that -value.

5. Decide if is in the domain of by checking if is defined (solid dot at ).

6. To determine continuity at , check if is defined, the two-sided limit exists, and equals the limit. If any of these fail, is not continuous at .

7. Repeat this process for each value of .

Try solving on your own before revealing the answer!