Business Calculus Midterm Review: Limits and Continuity from Graphs

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Q20. Use figure (1) to find the following limits using the graph of the function f(x):

Graph for Q20 (figure 1)

Background

Topic: Limits from Graphs

This question tests your ability to determine the value a function approaches as x approaches a specific value, using a graph. You are asked to find one-sided limits, which means you only consider the behavior of the function as x approaches from one side (left or right).

Key Terms and Concepts:

Limit: The value that a function approaches as the input approaches a certain value.
One-sided limit: The value the function approaches as x approaches from only one side (left or right ).

Step-by-Step Guidance

For each part, locate the relevant x-value on the graph (e.g., , , ).
For a right-hand limit (), trace the graph as x approaches a from values greater than a. For a left-hand limit (), trace from values less than a.
Observe the y-value that the function is approaching as x gets close to the target value from the specified side. Ignore the actual value of the function at that point (open or closed circle), and focus on the trend.
Write down the y-value the function is approaching for each limit, but do not state the final answer yet.

Try solving on your own before revealing the answer!

Final Answers:

For each, we looked at the y-value the function approaches from the specified side, regardless of whether the function is defined at that point.

Q21. Use figure (2) to find the following limits using the graph of the function f(x):

Graph for Q21 (figure 2)

Background

Topic: Limits from Graphs

This question asks you to determine the value a function approaches as x approaches a specific value, using a graph. Some limits may not exist if the function behaves differently from the left and right, or if the function is undefined in the interval.

Key Terms and Concepts:

Limit: The value a function approaches as x approaches a certain value.
Undefined interval: If the function is not defined for some x-values, the limit may not exist.

Step-by-Step Guidance

For each part, find the relevant x-value on the graph (, , ).
For two-sided limits, check the y-value the function approaches from both the left and right. If both sides approach the same value, the limit exists.
If the function is undefined at the point, but both sides approach the same value, the limit still exists.
If the function is undefined over an interval near the point, the limit may not exist. Check for gaps or jumps.

Try solving on your own before revealing the answer!

Final Answers:

a.
b.
c. does not exist (function is undefined over )

For each, we checked the behavior from both sides and noted if the function was undefined in the interval.

Q22. Use figure (3) to find the following limits using the graph of the function f(x):

Graph for Q22 (figure 3)

Background

Topic: One-Sided and Two-Sided Limits from Graphs

This question tests your ability to find one-sided limits and to recognize when a two-sided limit does not exist due to a jump or discontinuity.

Key Terms and Concepts:

One-sided limit: The value the function approaches from only one side.
Jump discontinuity: When the left and right limits at a point are not equal.

Step-by-Step Guidance

For each part, locate the x-value on the graph (, ).
For one-sided limits, trace the function as x approaches from the left or right and observe the y-value.
For two-sided limits, check if the left and right limits agree. If not, the limit does not exist.

Try solving on your own before revealing the answer!

Final Answers:

For (a) and (b), the left and right limits at are different, so the two-sided limit does not exist. For (c), the function approaches as approaches $2$.

Q23. Use figure (4) to find the following limits using the graph of the function f(x):

Graph for Q23 (figure 4)

Background

Topic: Limits Involving Infinite Behavior and Discontinuities

This question asks you to identify limits where the function may increase or decrease without bound (infinite limits), or where the function is discontinuous.

Key Terms and Concepts:

Infinite limit: The function increases or decreases without bound as x approaches a value.
Discontinuity: The function has a jump, hole, or asymptote at a point.

Step-by-Step Guidance

For each part, find the relevant x-value on the graph (, ).
For left-hand and right-hand limits, observe if the function goes to infinity () or negative infinity () as x approaches the value.
For two-sided limits, check if both sides approach the same value. If not, the limit does not exist.
For removable discontinuities, check the y-value the function approaches as x gets close to the point.

Try solving on your own before revealing the answer!

Final Answers:

a. (does not exist, but increases without bound)
b. does not exist (left and right sides do not agree)
c.

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