AP Calculus AB Quiz: Estimating Limit Values From Graphs

16 questions · exam conditions

0:00

Estimating Limit Values From GraphsQuestion 1 of 16

The graph of a function $f(x)$ has a vertical asymptote at the line $x=3$ . As $x$ approaches 3 from both the left and the right, the graph of $f(x)$ increases without bound.

Based on the description of the graph of $f(x)$ , what is $\lim_{x \to 3} f(x)$ ?

0

3

\infty

-\infty

← Back to quizzes

AP Calculus AB Quiz

AP Calculus AB Quiz: Estimating Limit Values From Graphs

Practice Estimating Limit Values From Graphs in AP Calculus AB 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 AB.

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

The graph of a function $f(x)$ has a vertical asymptote at the line $x=3$ . As $x$ approaches 3 from both the left and the right, the graph of $f(x)$ increases without bound.

Based on the description of the graph of $f(x)$ , what is $\lim_{x \to 3} f(x)$ ?

$0$
$3$
$\infty$ (correct answer)
$-\infty$

Explanation: The statement that the graph increases without bound as $x$ approaches 3 means that the function values are approaching positive infinity. Since this is the behavior from both sides, the limit is $\infty$ . While an infinite limit means the limit does not exist as a finite number, $\infty$ is the most precise description of the behavior.

Question 2

The graph of a function $h(x)$ has a vertical asymptote at $x=1$ and a removable discontinuity (hole) at $x=3$ . The graph indicates that $\lim_{x \to 3} h(x) = 5$ . The graph also shows that as $x$ approaches 1 from the right, the function decreases without bound.

Based on the description of the graph of $h(x)$ , what is the value of $\lim_{x \to 1^+} h(x)$ ?

$5$
$3$
$\infty$
$-\infty$ (correct answer)

Explanation: The question asks for the right-hand limit as $x$ approaches 1. The description states that the graph has a vertical asymptote at $x=1$ and that as $x$ approaches 1 from the right, the function decreases without bound. This means the function values approach negative infinity. The information about the removable discontinuity at $x=3$ is extraneous.

Question 3

The graph of $f(x)$ has a horizontal asymptote at $y=3$ as $x$ approaches infinity. The graph of $g(x)$ has a horizontal asymptote at $y=-1$ as $x$ approaches infinity.

Based on the descriptions of the graphs, what is $\lim_{x \to \infty} (f(x)g(x))$ ?

$2$
$-3$ (correct answer)
The limit does not exist.
$-1/3$

Explanation: Using the properties of limits, $\lim_{x \to \infty} (f(x)g(x)) = (\lim_{x \to \infty} f(x))(\lim_{x \to \infty} g(x))$ . From the descriptions, $\lim_{x \to \infty} f(x) = 3$ and $\lim_{x \to \infty} g(x) = -1$ . Therefore, the limit of the product is $(3)(-1) = -3$ .

Question 4

The graph of a function $f(x)$ is a smooth, continuous curve that passes through the point $(2, 5)$ .

Based on the description of the graph of $f(x)$ , what is the value of $\lim_{x \to 2} f(x)$ ?

$2$
$5$ (correct answer)
$0$
The limit does not exist.

Explanation: For a continuous function, the limit at a point is equal to the function's value at that point. Since the graph of $f(x)$ is continuous and passes through $(2, 5)$ , we have $f(2) = 5$ . Therefore, $\lim_{x \to 2} f(x) = f(2) = 5$ .

Question 5

The graph of a function $g(x)$ has a removable discontinuity at $x=3$ . As $x$ approaches 3 from either side, the $y$ -values on the graph approach 4. The function is defined at $x=3$ such that the point $(3, 1)$ is on the graph.

Based on the description of the graph of $g(x)$ , what is the value of $\lim_{x \to 3} g(x)$ ?

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

Explanation: The limit of a function as $x$ approaches a value is the $y$ -value that the function approaches. The description states that as $x$ approaches 3, the $y$ -values approach 4. The actual value of the function at $x=3$ , which is $g(3)=1$ , does not affect the limit.

Question 6

The graph of a function $f(x)$ has a jump discontinuity at $x=-2$ . The graph shows that as $x$ approaches -2 from the left side, the corresponding $y$ -values get closer and closer to 6. As $x$ approaches -2 from the right side, the $y$ -values get closer and closer to 0.

Based on the description of the graph of $f(x)$ , what is the value of $\lim_{x \to -2^-} f(x)$ ?

$6$ (correct answer)
$0$
The limit does not exist.
$-2$

Explanation: The notation $x \to -2^-$ indicates the limit as $x$ approaches -2 from the left. The description states that as $x$ approaches -2 from the left, the $y$ -values approach 6. Therefore, $\lim_{x \to -2^-} f(x) = 6$ .

Question 7

The graph of a function $g(x)$ is described as follows: for $x < 4$ , the graph is a parabola opening upwards with its vertex at $(2, 1)$ . For $x \ge 4$ , the graph is a horizontal line at $y=-3$ . The point $(4, -3)$ is on the graph.

Based on the description of the graph of $g(x)$ , what is the value of $\lim_{x \to 4^+} g(x)$ ?

$5$
$-3$ (correct answer)
The limit does not exist.
$1$

Explanation: The notation $x \to 4^+$ indicates the limit as $x$ approaches 4 from values greater than 4 (from the right). For $x > 4$ , the graph of $g(x)$ is the horizontal line $y=-3$ . Therefore, as $x$ approaches 4 from the right, the function values approach -3.

Question 8

A function $f(x)$ is defined on the closed interval $[0, 6]$ . The graph of the function begins at the point $(0, 2)$ and ends at the point $(6, -1)$ . The function is continuous on its domain.

Based on the description of the graph of $f(x)$ , what is the value of $\lim_{x \to 0} f(x)$ ?

$2$
$0$
The limit does not exist. (correct answer)
The limit cannot be determined.

Explanation: The two-sided limit $\lim_{x \to 0} f(x)$ does not exist because the function is not defined for $x < 0$ . Therefore, the left-hand limit $\lim_{x \to 0^-} f(x)$ does not exist. For the two-sided limit to exist, both one-sided limits must exist and be equal. Note that the one-sided limit from the right, $\lim_{x \to 0^+} f(x)$ , does exist and is equal to 2.

Question 9

The graph of $f(x)$ is a continuous curve that passes through the point $(-2, 4)$ . The graph of $g(x)$ has a removable discontinuity at $x=-2$ , but as $x$ approaches -2, the graph of $g(x)$ approaches $y=1$ .

Based on the descriptions of the graphs, what is $\lim_{x \to -2} (f(x) - g(x))$ ?

$3$ (correct answer)
$5$
The limit does not exist.
The limit cannot be determined.

Explanation: Using the properties of limits, $\lim_{x \to -2} (f(x) - g(x)) = \lim_{x \to -2} f(x) - \lim_{x \to -2} g(x)$ . From the descriptions, $\lim_{x \to -2} f(x) = 4$ and $\lim_{x \to -2} g(x) = 1$ . Therefore, the limit is $4 - 1 = 3$ .

Question 10

The graph of a function $f(x)$ is a horizontal line at $y=8$ . The graph of a function $g(x)$ is a continuous curve that passes through the point $(5, 2)$ .

Based on the descriptions of the graphs, what is $\lim_{x \to 5} \frac{f(x)}{g(x)}$ ?

$10$
$6$
$4$ (correct answer)
The limit does not exist.

Explanation: Using the properties of limits, $\lim_{x \to 5} \frac{f(x)}{g(x)} = \frac{\lim_{x \to 5} f(x)}{\lim_{x \to 5} g(x)}$ . From the descriptions, $\lim_{x \to 5} f(x) = 8$ and $\lim_{x \to 5} g(x) = 2$ . Therefore, the limit of the quotient is $\frac{8}{2} = 4$ .

Question 11

The graph of a function $g(x)$ has a removable discontinuity at $x=2$ , and $\lim_{x \to 2} g(x) = -1$ . The graph of a function $f(x)$ is continuous at $x=-1$ and passes through the point $(-1, 5)$ .

Based on the descriptions of the graphs, what is the value of $\lim_{x \to 2} f(g(x))$ ?

$-1$
$2$
$5$ (correct answer)
The limit does not exist.

Explanation: To evaluate the limit of a composite function, we first find the limit of the inner function: $\lim_{x \to 2} g(x) = -1$ . Let $u = g(x)$ ; as $x \to 2$ , $u \to -1$ . The problem becomes finding $\lim_{u \to -1} f(u)$ . Since $f(x)$ is continuous at $x=-1$ , this limit is $f(-1)$ , which is 5.

Question 12

The graph of the function $f(x)$ has a horizontal asymptote at the line $y=4$ as $x$ approaches positive infinity.

From the graphical behavior described, what is the value of $\lim_{x \to \infty} f(x)$ ?

$0$
$4$ (correct answer)
$\infty$
The limit cannot be determined.

Explanation: A horizontal asymptote of a graph of a function describes the end behavior of the function. The statement that $y=4$ is a horizontal asymptote as $x \to \infty$ means that the function values $f(x)$ approach 4 as $x$ increases without bound. This is the definition of $\lim_{x \to \infty} f(x) = 4$ .

Question 13

The graph of a function $h(x)$ has two distinct horizontal asymptotes. The graph approaches the line $y=6$ as $x$ grows infinitely large in the positive direction, and it approaches the line $y=-2$ as $x$ grows infinitely large in the negative direction.

Based on the description of the graph of $h(x)$ , what is the value of $\lim_{x \to -\infty} h(x)$ ?

$6$
$-2$ (correct answer)
$\infty$
The limit does not exist.

Explanation: The limit as $x \to -\infty$ describes the behavior of the function as $x$ moves to the far left. The description states that the graph approaches the line $y=-2$ as $x$ grows infinitely large in the negative direction. Therefore, $\lim_{x \to -\infty} h(x) = -2$ .

Question 14

The graph of a function $f(x)$ has a removable discontinuity at $x=0$ . As $x$ approaches 0 from the left, $f(x)$ approaches 3. As $x$ approaches 0 from the right, $f(x)$ also approaches 3. However, the function is defined as $f(0)=5$ .

Based on the description of the graph of $f(x)$ , what is the value of $\lim_{x \to 0} f(x)$ ?

$3$ (correct answer)
$5$
The limit does not exist.
The limit is undefined.

Explanation: The limit of a function at a point depends on the value the function approaches from both sides, not the actual value at the point. Since both the left-hand limit ( $\lim_{x \to 0^-} f(x) = 3$ ) and the right-hand limit ( $\lim_{x \to 0^+} f(x) = 3$ ) are equal, the two-sided limit exists and equals 3. This describes a removable discontinuity because the limit exists but differs from the function value at that point.

Question 15

The graph of a function $f(x)$ has a removable discontinuity at $x=c$ where the limit is $L$ . The graph also has a jump discontinuity at $x=d$ where the left-hand limit is $M$ and the right-hand limit is $N$ ( $M \neq N$ ).

Which of the following must be true about the graph of $f(x)$ ?

$f(c)$ must be equal to $L$ .
$\lim_{x \to c} f(x)$ does not exist.
$\lim_{x \to d} f(x)$ does not exist. (correct answer)
$f(d)$ must be equal to either $M$ or $N$ .

Explanation: At a jump discontinuity, the left-hand and right-hand limits are not equal. Since $\lim_{x \to d^-} f(x) = M$ and $\lim_{x \to d^+} f(x) = N$ with $M \neq N$ , the two-sided limit $\lim_{x \to d} f(x)$ does not exist. For a removable discontinuity at $x=c$ , the limit $L$ exists, but it is not necessarily equal to $f(c)$ . The value of $f(d)$ could be anything and is not restricted to be $M$ or $N$ .

Question 16

The graph of a function $g(x)$ is continuous for all real numbers. The graph passes through the origin $(0,0)$ , has a local maximum at $(2,4)$ , and a local minimum at $(5,1)$ .

Based on the description of the graph of $g(x)$ , what is $\lim_{x \to 2} g(x)$ ?

$0$
$1$
$2$
$4$ (correct answer)

Explanation: The problem states that the graph of $g(x)$ is continuous for all real numbers. For a continuous function, the limit at any point is equal to the function's value at that point. Since the graph has a local maximum at $(2,4)$ , the point $(2,4)$ is on the graph, meaning $g(2)=4$ . Therefore, $\lim_{x \to 2} g(x) = g(2) = 4$ .