Q1. Find using the graph.

Background

Topic: One-Sided Limits from a Graph

This question tests your ability to determine the left-hand limit of a function at a specific point by analyzing its graph. The left-hand limit, , is the value that approaches as approaches from the left.

Key Terms and Formulas:

• Left-hand limit:

• Graphical approach: Trace the curve as approaches from values less than $a$.

Step-by-Step Guidance

1. Locate on the -axis of the graph.

2. Observe the behavior of as approaches from the left (values less than $-1$).

3. Follow the curve from the left side and see what -value the function is approaching as gets closer to .

4. Identify if the function approaches a specific -value, or if there is a jump, hole, or asymptote at .

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Q2. Find using the graph.

Background

Topic: One-Sided Limits from a Graph

This question asks you to determine the right-hand limit of a function at a specific point by analyzing its graph. The right-hand limit, , is the value that approaches as approaches from the right.

Key Terms and Formulas:

• Right-hand limit:

• Graphical approach: Trace the curve as approaches from values greater than $a$.

Step-by-Step Guidance

1. Find on the -axis of the graph.

2. Look at the behavior of as approaches from the right (values greater than $-1$).

3. Follow the curve from the right side and see what -value the function is approaching as gets closer to .

4. Check if the function approaches a specific -value, or if there is a discontinuity at .

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Q3. What is ?

Background

Topic: Evaluating a Function from a Graph

This question asks you to find the value of the function at a specific point by looking at the graph. is the -value of the function at .

Key Terms and Formulas:

• Function value: is the -coordinate of the point on the graph where .

• Closed circle: Indicates the function is defined at that point.

• Open circle: Indicates the function is not defined at that point.

Step-by-Step Guidance

1. Locate on the -axis of the graph.

2. Look for a closed (filled-in) circle at to determine the value of .

3. If there is a closed circle, the -value at that point is .

4. If there is an open circle, is not defined at that -value; check for a closed circle at a different $y$-value at .

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Q4. Find using the graph.

Background

Topic: Two-Sided Limits from a Graph

This question tests your understanding of the two-sided limit at a point. The two-sided limit exists only if both the left-hand and right-hand limits exist and are equal.

Key Terms and Formulas:

• Two-sided limit:

• Exists if:

Step-by-Step Guidance

1. Recall the left-hand and right-hand limits you found for .

2. Compare the values of and .

3. If the two one-sided limits are not equal, the two-sided limit does not exist at .

4. If they are equal, the two-sided limit is that common value.

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Q5. Is continuous at ?

Background

Topic: Continuity from a Graph

This question tests your understanding of the definition of continuity at a point. A function is continuous at if the following three conditions are met:

• is defined

• exists

Key Terms and Formulas:

• Continuity at : is defined, exists, and

Step-by-Step Guidance

1. Check if is defined by looking for a closed circle at .

2. Determine if exists by comparing the left and right limits.

3. If both are true, check if the limit equals the function value at .

4. If any of these conditions fail, the function is not continuous at .

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