Limits and Continuity from Graphs – Business Calculus Study Guidance

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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 the behavior of a function as x approaches a specific value, using a graph. You must distinguish between the value the function approaches (the limit) and the actual value of the function at that point.

Graph of function for Q20

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 dots: Open dots indicate the function is not defined at that point; closed dots indicate the function is defined there.

Step-by-Step Guidance

For each part, identify the x-value you are approaching and whether it is from the left () or right ().
Examine the graph near the specified x-value. Observe the y-values the function approaches as x gets closer to the target from the correct side.
For part (a), look at the behavior as x approaches from the right. For part (b), approach $2-1$ from the right.
Note whether the function is defined at the point (open or closed dot), but remember the limit depends on the approach, not the actual value at the point.
Write down the y-value the function is approaching for each part, but do not compute the final answer yet. If the function jumps or is undefined, consider whether the limit exists.

Try solving on your own before revealing the answer!

Final Answers:

For each, the limit is the y-value the function approaches from the specified side, 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 determine the value a function approaches as x gets close to a specific value, using a graph. Pay attention to whether the function is defined at the point and whether the limit exists from both sides.

Graph of function for Q21

Key Terms:

Limit: The value a function approaches as x approaches a certain value.
Undefined: The function may not be defined at the point, but the limit can still exist.
Interval notation: Used to describe where the function is defined or undefined.

Step-by-Step Guidance

For each part, identify the x-value you are approaching.
Examine the graph near the specified x-value. Observe the y-values the function approaches as x gets closer to the target from both sides.
For part (a), approach from both sides. For part (b), approach from both sides. For part (c), approach from both sides.
Check if the function is defined at the point, but focus on the behavior as x approaches the value.
If the function is undefined or the left/right limits do not agree, consider whether the limit exists.

Try solving on your own before revealing the answer!

Final Answers:

a.
b.
c. does not exist

For (c), the function is undefined over [1, 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. If the left and right limits are not equal, the general limit does not exist at that point.

Graph of function for Q22

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 part (a), approach from the left; for part (b), approach $x = -3$ from the right; for part (c), approach from both sides.
Observe the y-values the function approaches as x gets close to the target from the specified side.
If the left and right limits are not equal, the general limit does not exist at that point.
For part (c), check if the function is defined at , but focus on the value the function approaches as x gets close to 2.

Try solving on your own before revealing the answer!

Final Answers:

For (a) and (b), the left and right limits are not equal, so the general limit at does not exist. For (c), the function approaches as x approaches 2.

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 when a function approaches infinity or does not exist as x approaches a value, using a graph. Pay attention to vertical asymptotes and jumps.

Graph of function for Q23

Key Terms:

Vertical asymptote: A line where the function increases or decreases without bound.
Infinite limit: The function approaches or as x approaches a value.
Discontinuity: The function may jump or be undefined at certain points.

Step-by-Step Guidance

For part (a), approach from the left; for part (b), approach $x = 4$ from both sides; for part (c), approach from both sides.
Observe the behavior of the function as x gets close to the target value. If the function increases or decreases without bound, the limit is infinite.
If the left and right limits do not agree, the general limit does not exist.
For part (c), check the value the function approaches as x gets close to 2.

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 (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.