Business Calculus Midterm Review: Limits and Continuity from Graphs

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Q20. Using figure (1), find the limit of f(x) as x approaches the following values:

Background

Topic: Limits from Graphs

This question tests your ability to interpret one-sided and two-sided limits using a graph. You must observe the behavior of the function as x approaches a specific value from the left or right.

Graph for figure (1)

Key Terms:

Limit: The value a function approaches as the input approaches a certain point.
One-sided limit: The value approached from only one side (left or right).
Open/closed circles: Indicate whether the function is defined at a point.

Step-by-Step Guidance

For each part, identify the direction from which x is approaching (left or right).
Examine the graph near the specified x-value. Look at the y-values the function approaches as x gets closer to the target from the correct side.
For open circles, remember the function may not be defined at that point, but the limit can still exist if the graph approaches a specific value.
For each part, write down the y-value the function is approaching, but do not state the final answer yet.
Set up your answer for each limit based on your observations, but leave the final value for you to determine.

Try solving on your own before revealing the answer!

Final Answers:

Each answer is based on the y-value the function approaches from the specified direction, regardless of whether the function is defined at that point.

Q21. Using figure (2), find the limit of f(x) as x approaches the following values:

Background

Topic: Limits from Graphs

This question tests your ability to find limits using a graph, including cases where the function is undefined at the point but the limit exists.

Graph for figure (2)

Key Terms:

Limit at a point: The value the function approaches as x gets close to a specific value.
Undefined points: The function may not be defined at the point, but the limit can still exist.

Step-by-Step Guidance

For each part, observe the graph as x approaches the specified value from both sides.
Check if the function values approach a single y-value, even if the function is not defined at that point.
For part c, check if the function is defined over the interval near x = 2.
Write down the y-value the function approaches for each part, but do not state the final answer yet.
Set up your answer for each limit based on your observations, but leave the final value for you to determine.

Try solving on your own before revealing the answer!

Final Answers:

a.
b.
c. does not exist

For part c, the function is undefined over the interval near x = 2, so the limit does not exist.

Q22. Using figure (3), find the limit of f(x) as x approaches the following values:

Background

Topic: One-sided Limits from Graphs

This question tests your ability to find left-hand and right-hand limits using a graph, and to recognize when the two-sided limit does not exist.

Graph for figure (3)

Key Terms:

Left-hand limit: The value the function approaches as x approaches from the left.
Right-hand limit: The value the function approaches as x approaches from the right.
Discontinuity: Occurs when left and right limits are not equal.

Step-by-Step Guidance

For each part, identify whether you are approaching from the left or right.
Observe the graph near the specified x-value and note the y-value the function approaches.
For part a and b, compare the left and right limits to see if the two-sided limit exists.
For part c, check the behavior as x approaches 2.
Set up your answer for each limit based on your observations, but leave the final value for you to determine.

Try solving on your own before revealing the answer!

Final Answers:

For part a and b, the left and right limits are not equal, so the two-sided limit does not exist at x = -3.

Q23. Using figure (4), find the limit of f(x) as x approaches the following values:

Background

Topic: Limits and Infinite Behavior from Graphs

This question tests your ability to recognize infinite limits and discontinuities using a graph.

Graph for figure (4)

Key Terms:

Infinite limit: The function increases or decreases without bound as x approaches a value.
Discontinuity: The function is not defined or the limit does not exist at a point.

Step-by-Step Guidance

For part a, observe the graph as x approaches 4 from the left. Note if the function increases without bound.
For part b, check if the left and right limits agree or if the function is defined at x = 4.
For part c, observe the behavior as x approaches 2 and note the y-value the function approaches.
Set up your answer for each limit based on your observations, but leave the final value for you to determine.

Try solving on your own before revealing the answer!

Final Answers:

a. (does not exist, but increases without bound)
b. does not exist
c.

For part a, the function increases without bound as x approaches 4 from the left. For part b, the left and right limits do not agree, so the limit does not exist. For part c, the function approaches 3 as x approaches 2.