IndietroLimits and Continuity from Graphs – Business Calculus Study Guidance
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Q20. Using figure (1), find the limit of each function f as specified:
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 and the value at the point itself.
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).
Discontinuity: A point where the function is not continuous.
Step-by-Step Guidance
Examine figure (1) for the behavior of the function near the specified x-values.
For each part, focus on the direction specified (left or right) and observe the y-values the function approaches as x gets close to the target value.
For part (a), look at the function as x approaches -7 from the right. Identify the y-value the curve approaches, not the value at x = -7.
For part (b), observe the function as x approaches 2 from the left. Determine the y-value the curve approaches from that side.
For part (c), consider the function as x approaches -1 from the right. Find the y-value the curve approaches from that direction.
Check for any open or closed circles at these points, as they indicate whether the function is defined at the point or not, but remember the limit depends on the approach, not the actual value at the point.
Try solving on your own before revealing the answer!
Final Answers:
a.
b.
c.
For each part, the limit is 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 each function f as specified:
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.
Key Terms:
Limit: The value a function approaches as x approaches a certain point.
Undefined: The function may not be defined at the point, but the limit can still exist.
Does not exist (DNE): The limit may not exist if the function behaves differently from each side or is undefined over an interval.
Step-by-Step Guidance
Examine figure (2) for the behavior of the function near the specified x-values.
For part (a), observe the function as x approaches 0 from both sides. Identify the y-value the curve approaches.
For part (b), look at the function as x approaches -8 from both sides. Determine the y-value the curve approaches.
For part (c), consider the function as x approaches 2. Check if the function is defined over the interval near 2 and whether the left and right limits agree.
Check for open circles or gaps in the graph, as these indicate discontinuities or undefined regions.
Try solving on your own before revealing the answer!
Final Answers:
a.
b.
c. does not exist
The limit exists for parts (a) and (b) because the function approaches a single value from both sides. For part (c), the function is undefined over [1, 2), so the limit does not exist.
Q22. Using figure (3), find the limit of each function f as specified:
Background
Topic: Limits from Graphs
This question tests your ability to find left-hand and right-hand limits, and to recognize when the limit does not exist due to a jump or discontinuity.
Key Terms:
Left-hand limit: The value the function approaches as x comes from the left.
Right-hand limit: The value the function approaches as x comes from the right.
Jump discontinuity: When the left and right limits at a point are not equal.
Step-by-Step Guidance
Examine figure (3) for the behavior of the function near the specified x-values.
For part (a), observe the function as x approaches -3 from the left. Identify the y-value the curve approaches.
For part (b), look at the function as x approaches -3 from the right. Determine the y-value the curve approaches.
For part (c), consider the function as x approaches 2. Check the y-value the curve approaches, even if the function is undefined at x = 2.
Compare the left and right limits at -3 to see if the overall limit exists.
Try solving on your own before revealing the answer!
Final Answers:
a.
b.
c.
At x = -3, the left and right limits are not equal, so the overall limit does not exist. At x = 2, the function approaches -4 even if it is undefined at that point.
Q23. Using figure (4), find the limit of each function f as specified:
Background
Topic: Limits from Graphs
This question tests your ability to recognize infinite limits and discontinuities from a graph, and to distinguish between one-sided and two-sided limits.
Key Terms:
Infinite limit: The function increases or decreases without bound as x approaches a point.
Does not exist (DNE): The limit does not exist if the function behaves differently from each side.
One-sided limit: The value approached from only one side.
Step-by-Step Guidance
Examine figure (4) for the behavior of the function near the specified x-values.
For part (a), observe the function as x approaches 4 from the left. Determine if the function increases without bound.
For part (b), look at the function as x approaches 4 from both sides. Check if the left and right limits agree or if the function is undefined.
For part (c), consider the function as x approaches 2. Identify the y-value the curve approaches, regardless of the value at x = 2.
Check for vertical asymptotes or open circles, which indicate infinite limits or discontinuities.
Try solving on your own before revealing the answer!
Final Answers:
a. (DNE)
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.
