AP Calculus BC Quiz: Estimating Limit Values From Graphs

What this quiz covers

This quiz focuses on Estimating Limit Values From Graphs, giving you a quick way to practice the rules, question types, and explanations that matter most for AP Calculus BC.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

Question 1

1. 5 (correct answer)
2. 3
3. 2
4. The limit does not exist

Explanation: The correct answer is A. The limit of a function as x approaches a value depends only on the behavior of the function near that point, not the actual function value at that point. From the graph, as x approaches 2 from both the left and right sides, the y-values approach 5. The fact that there's a hole at (2,5) and f(2) = 3 doesn't affect the limit value.

Question 2

1. -3, matching the behavior from the left side
2. The limit approaches $-\infty$ from the right
3. The right-hand limit does not exist (correct answer)
4. 1, since that's the x-value being approached

Explanation: The correct answer is C. The notation $\lim_{x \to 1^+} s(x)$ asks specifically for the right-hand limit. Since the function decreases without bound (approaches negative infinity) as x approaches 1 from the right, the right-hand limit does not exist. While we can describe this behavior as approaching negative infinity, limits that approach infinity are considered to not exist in the traditional sense.

Question 3

1. The limit is exactly 1.5 based on the visual evidence
2. The limit is approximately 1.5, within the uncertainty of graphical reading (correct answer)
3. The limit cannot be determined due to insufficient graph resolution
4. The limit should be rounded to 2 for simplicity purposes

Explanation: The correct answer is B. Graphical limit estimation inherently involves some uncertainty due to scale and resolution limitations. The best we can do is provide a reasonable approximation based on visual evidence. Saying 'approximately 1.5' acknowledges both the apparent trend and the inherent limitations of graphical analysis.

Question 4

1. $+\infty$, since both sides approach positive infinity
2. 7, since that's the location of the asymptote
3. The limit does not exist due to infinite behavior (correct answer)
4. 0, representing the reciprocal of the infinite behavior

Explanation: The correct answer is C. Even though both one-sided limits have the same infinite behavior (both approach +∞), the limit does not exist in the formal sense because infinity is not a real number. We can describe the behavior as approaching positive infinity, but technically this means the limit does not exist.

Question 5

1. 3, since the dampening oscillation settles toward this value (correct answer)
2. 2, since that's the x-value being approached
3. The limit does not exist due to the oscillatory behavior
4. 0, representing the final amplitude of the dampened oscillation

Explanation: The correct answer is A. Dampened oscillation that settles toward a specific value indicates that the limit exists and equals that settling value. Unlike persistent oscillation (which prevents limits from existing), dampened oscillation that converges to a value allows the limit to exist at that converged value.

Question 6

1. 4, since that's the actual function value at x = 8
2. 0, since that's where both sides of the curve approach (correct answer)
3. 2, representing the average of the approaches and function value
4. 8, since that's the x-coordinate of the point in question

Explanation: The correct answer is B. The limit is determined by the approaching behavior of the function, not its actual value at the point. Since both one-sided limits equal 0, the two-sided limit is 0, even though b(8) = 4. This represents a point discontinuity.

Question 7

1. -2, since both sides approach this value despite the gap (correct answer)
2. -5, representing the x-coordinate where the gap occurs
3. The limit does not exist because of the gap
4. 0, since gaps typically indicate zero-valued behavior

Explanation: The correct answer is A. A gap in the function (removable discontinuity) doesn't prevent a limit from existing if both one-sided limits exist and are equal. Since the function approaches -2 from both sides, the limit equals -2, regardless of what happens exactly at x = -5.

Question 8

1. 3, since the curve approaches the location of the hole (correct answer)
2. 5, representing the x-coordinate of the hole location
3. Undefined, since the function has a hole at x = 5
4. 0, since holes typically represent absence of value

Explanation: The correct answer is A. A hole (removable discontinuity) shows where the function would naturally go based on the surrounding curve behavior. Even though f(5) is undefined, the limit exists and equals the y-coordinate of the hole (3) because both one-sided limits approach this value.

Question 9

1. 5, since both pieces approach the same y-value (correct answer)
2. -3, representing the x-coordinate of the gap location
3. The limit does not exist because there's a gap
4. Undefined since the function has no value at x = -3

Explanation: The correct answer is A. Despite the gap at x = -3, both sides of the function approach the same y-value (5). The open circles indicate that the function isn't defined at x = -3, but this doesn't prevent the limit from existing. Since both one-sided limits equal 5, the two-sided limit is 5.

Question 10

1. 0, since we're taking the limit as x approaches zero
2. 4, since the function value and limit both equal 4 (correct answer)
3. Undefined, since we cannot determine limits at defined points
4. The limit requires additional information about the function's behavior

Explanation: The correct answer is B. When a function is continuous at a point (no breaks, holes, or jumps), the limit as x approaches that point equals the function value at that point. Since the graph shows t(x) is continuous at x = 0 and t(0) = 4, we have lim(x→0) t(x) = 4.

Question 11

1. 1, since that's where the function is actually defined
2. 6, since that's where the curve approaches the open circle (correct answer)
3. The limit cannot exist when there's a discontinuity present
4. 3.5, representing the midpoint between the two marked values

Explanation: The correct answer is B. The limit depends on where the function approaches, not where it's actually defined. The open circle at (-2, 6) indicates where the function would naturally go based on the surrounding curve behavior, while the solid dot at (-2, 1) shows an artificially defined value. The limit is 6.

Question 12

1. 6, representing the left-hand limit specifically (correct answer)
2. 10, representing the right-hand limit value
3. 8, representing the average of both one-sided limits
4. The limit does not exist due to the jump discontinuity

Explanation: The correct answer is A. The notation lim(x→9⁻) specifically asks for the left-hand limit, which is the value the function approaches as x approaches 9 from the left side only. From the graph, this value is 6. The jump discontinuity affects the two-sided limit but not the one-sided limits individually.

Question 13

1. Exactly 3.2 since that's what the graph clearly shows
2. Approximately 3.2, recognizing this is a graphical estimate (correct answer)
3. 3 since we should round to the nearest integer value
4. Cannot be determined without the algebraic function expression

Explanation: The correct answer is B. When estimating limits from graphs, we must recognize that our answers are approximations based on visual inspection. The scale and resolution of the graph affect our precision, so 'approximately 3.2' is the most honest assessment of what can be determined graphically.

Question 14

1. 2, since both sides approach the same value at the cusp (correct answer)
2. The limit does not exist because there's a sharp corner
3. 3, since that's the x-coordinate of the cusp point
4. Undefined due to the non-smooth behavior at that point

Explanation: The correct answer is A. A cusp or sharp corner affects differentiability but not the existence of limits. Since the function approaches the same y-value (2) from both sides as x approaches 3, the limit exists and equals 2. Cusps create non-differentiable points but don't prevent limits from existing.

Question 15

1. 8, since both one-sided limits equal 8 (correct answer)
2. Undefined, since v(6) is not defined
3. 6, representing the x-value of the approaching point
4. The limit cannot exist when the function is undefined

Explanation: The correct answer is A. Limits describe the behavior of a function as it approaches a point, not what happens at the point itself. Since both one-sided limits equal 8, the two-sided limit equals 8, regardless of whether the function is defined at x = 6.

Question 16

1. The limit equals 7 since that's the left-hand approach value
2. The limit equals 2 since that's the right-hand approach value
3. The limit equals 4.5, which is the average of the approach values
4. The limit does not exist since the one-sided limits are different (correct answer)

Explanation: The correct answer is D. For a two-sided limit to exist, both one-sided limits must exist and be equal. Since the left-hand limit (7) and right-hand limit (2) are different, the two-sided limit does not exist. This represents a jump discontinuity in the function.

Question 17

1. 1.2, since the function levels off at this horizontal value (correct answer)
2. 10, since that's the x-value being approached
3. 0, representing the slope of the horizontal leveling behavior
4. The limit does not exist due to horizontal asymptote behavior

Explanation: The correct answer is A. When a graph clearly shows the function leveling off horizontally at a specific y-value as x approaches a point, this indicates the limit equals that y-value. The horizontal leveling at y = 1.2 from both sides indicates that lim(x→10) e(x) = 1.2.

Question 18

1. $-1$
2. $0$
3. $1$ (correct answer)
4. $2$
5. DNE

Explanation: To estimate this two-sided limit, we need to examine how the function behaves as x approaches 0 from both directions. Tracing along the curve from the left (negative x-values moving right) and from the right (positive x-values moving left), we see that both paths lead the function values toward y = 1. Since the left-hand and right-hand approaches agree at y = 1, the two-sided limit exists and equals 1. The actual function value at x = 0 might be different or undefined, but this doesn't affect the limit. Students often confuse the limit with the function value, but remember that limits describe approach behavior, not arrival values. For two-sided limits to exist, always verify that both one-sided limits exist and are equal.