AP Calculus BC Quiz: Estimating Limit Values From Graphs

18 questions · exam conditions

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Estimating Limit Values From GraphsQuestion 1 of 18

Based on the graph of function $f(x)$ , what is $\lim_{x \to 2} f(x)$ ? The graph shows that as $x$ approaches 2 from both sides, the function values approach 5, even though there is a hole at the point $(2, 5)$ and the function is actually defined as $f(2) = 3$ .

The limit does not exist

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AP Calculus BC Quiz

AP Calculus BC Quiz: Estimating Limit Values From Graphs

Practice Estimating Limit Values From Graphs in AP Calculus BC with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.

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.

All questions

Question 1

5 (correct answer)
3
2
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

From the graph of $s(x)$ , as $x$ approaches 1 from the left, $s(x)$ approaches -3. As $x$ approaches 1 from the right, $s(x)$ decreases without bound toward negative infinity. What is $\lim_{x \to 1^+} s(x)$ ?

-3, matching the behavior from the left side
The limit approaches $-\infty$ from the right
The right-hand limit does not exist (correct answer)
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

From the graph of $w(x)$ , as $x$ approaches -4, the function values appear to approach 1.5, but the resolution of the graph makes it difficult to be completely certain. The y-values seem to be between 1.4 and 1.6. What can be concluded about $\lim_{x \to -4} w(x)$ ?

The limit is exactly 1.5 based on the visual evidence
The limit is approximately 1.5, within the uncertainty of graphical reading (correct answer)
The limit cannot be determined due to insufficient graph resolution
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

The graph shows $y(x)$ with a vertical asymptote at $x = 7$ . As $x$ approaches 7 from the left, $y(x)$ increases without bound, and as $x$ approaches 7 from the right, $y(x)$ also increases without bound. What is $\lim_{x \to 7} y(x)$ ?

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

The graph of $a(x)$ exhibits a periodic oscillation that dampens as $x$ approaches 2. The amplitude of oscillation decreases, and the function values appear to settle toward 3 as $x$ gets closer to 2. What is $\lim_{x \to 2} a(x)$ ?

3, since the dampening oscillation settles toward this value (correct answer)
2, since that's the x-value being approached
The limit does not exist due to the oscillatory behavior
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

From the graph of $b(x)$ , as $x$ approaches 8 from the left, $b(x)$ approaches 0, and as $x$ approaches 8 from the right, $b(x)$ approaches 0. The function has $b(8) = 4$ marked with a solid dot. What is $\lim_{x \to 8} b(x)$ ?

4, since that's the actual function value at x = 8
0, since that's where both sides of the curve approach (correct answer)
2, representing the average of the approaches and function value
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

The graph of $c(x)$ shows a smooth, continuous curve everywhere except at $x = -5$ , where there's a small gap. As $x$ approaches -5 from both directions, the function values approach -2. What is $\lim_{x \to -5} c(x)$ ?

-2, since both sides approach this value despite the gap (correct answer)
-5, representing the x-coordinate where the gap occurs
The limit does not exist because of the gap
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

From the graph of $f(x)$ , there's a hole at $(5, 3)$ indicated by an open circle. The surrounding curve behavior shows that as $x$ approaches 5 from either side, $f(x)$ approaches 3. The function is undefined at $x = 5$ . What is $\lim_{x \to 5} f(x)$ ?

3, since the curve approaches the location of the hole (correct answer)
5, representing the x-coordinate of the hole location
Undefined, since the function has a hole at x = 5
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

Looking at the graph of $z(x)$ , there's a gap in the curve at $x = -3$ . The left piece of the curve ends at $(-3, 5)$ with an open circle, and the right piece begins at $(-3, 5)$ with an open circle. What is $\lim_{x \to -3} z(x)$ ?

5, since both pieces approach the same y-value (correct answer)
-3, representing the x-coordinate of the gap location
The limit does not exist because there's a gap
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

The graph of $t(x)$ shows a smooth curve that passes through the point $(0, 4)$ without any breaks, holes, or unusual behavior near $x = 0$ . What is $\lim_{x \to 0} t(x)$ ?

0, since we're taking the limit as x approaches zero
4, since the function value and limit both equal 4 (correct answer)
Undefined, since we cannot determine limits at defined points
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

The graph shows function $r(x)$ with a removable discontinuity at $x = -2$ . There's an open circle at $(-2, 6)$ and a solid dot at $(-2, 1)$ . As $x$ approaches -2 from both sides, the curve heads toward the open circle. What is $\lim_{x \to -2} r(x)$ ?

1, since that's where the function is actually defined
6, since that's where the curve approaches the open circle (correct answer)
The limit cannot exist when there's a discontinuity present
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

Looking at the graph of $d(x)$ , as $x$ approaches 9 from the left, $d(x)$ approaches 6. As $x$ approaches 9 from the right, $d(x)$ approaches 10. There's a jump discontinuity at $x = 9$ . What is $\lim_{x \to 9^-} d(x)$ ?

6, representing the left-hand limit specifically (correct answer)
10, representing the right-hand limit value
8, representing the average of both one-sided limits
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

From the graph of function $q(x)$ , it appears that as $x$ approaches 5 from both sides, $q(x)$ approaches 3.2. However, due to the scale of the graph, this is an approximation. What is the best estimate for $\lim_{x \to 5} q(x)$ ?

Exactly 3.2 since that's what the graph clearly shows
Approximately 3.2, recognizing this is a graphical estimate (correct answer)
3 since we should round to the nearest integer value
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

Looking at the graph of $u(x)$ , there's a sharp corner (cusp) at $x = 3$ where the function value is 2. The function approaches this point from both sides, reaching the same y-value. What is $\lim_{x \to 3} u(x)$ ?

2, since both sides approach the same value at the cusp (correct answer)
The limit does not exist because there's a sharp corner
3, since that's the x-coordinate of the cusp point
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

The graph of $v(x)$ shows that as $x$ approaches 6 from the left, $v(x)$ approaches 8, and as $x$ approaches 6 from the right, $v(x)$ approaches 8. However, $v(6)$ is undefined (there's no point plotted at $x = 6$ ). What is $\lim_{x \to 6} v(x)$ ?

8, since both one-sided limits equal 8 (correct answer)
Undefined, since v(6) is not defined
6, representing the x-value of the approaching point
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

From the graph of function $h(x)$ , as $x$ approaches 3 from the left, $h(x)$ approaches 7, but as $x$ approaches 3 from the right, $h(x)$ approaches 2. What can be concluded about $\lim_{x \to 3} h(x)$ ?

The limit equals 7 since that's the left-hand approach value
The limit equals 2 since that's the right-hand approach value
The limit equals 4.5, which is the average of the approach values
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

The graph of $e(x)$ shows that as $x$ approaches 10, the function values get arbitrarily close to 1.2. The curve appears to level off horizontally near $y = 1.2$ as it approaches $x = 10$ from both sides. What is $\lim_{x \to 10} e(x)$ ?

1.2, since the function levels off at this horizontal value (correct answer)
10, since that's the x-value being approached
0, representing the slope of the horizontal leveling behavior
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

For the graphed function $p$ , estimate $\lim_{x\to 0} p(x)$ .

$-1$
$0$
$1$ (correct answer)
$2$
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.