AP Calculus AB Quiz: Estimating Limit Values From Tables

19 questions · exam conditions

0:00

Estimating Limit Values From TablesQuestion 1 of 19

The table below gives values of function $r(x)$ near $x = 5$ :

$x$	4.8	4.9	4.99	5.01	5.1	5.2
$r(x)$	23.04	24.01	24.9801	25.0201	26.01	27.04

The best estimate for $\lim_{x \to 5} r(x)$ is:

25

based on the convergence pattern observed

5

since that's the value

x

approaches

24.9801

which is the closest left-side value

25.0201

which is the closest right-side value

← Back to quizzes

AP Calculus AB Quiz

AP Calculus AB Quiz: Estimating Limit Values From Tables

Practice Estimating Limit Values From Tables 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 Tables, 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 table below gives values of function $r(x)$ near $x = 5$ :

$x$	4.8	4.9	4.99	5.01	5.1	5.2
$r(x)$	23.04	24.01	24.9801	25.0201	26.01	27.04

The best estimate for $\lim_{x \to 5} r(x)$ is:

$25$ based on the convergence pattern observed (correct answer)
$5$ since that's the value $x$ approaches
$24.9801$ which is the closest left-side value
$25.0201$ which is the closest right-side value

Explanation: The table shows values approaching 25 from both sides: from the left (23.04 → 24.01 → 24.9801) and from the right (25.0201 → 26.01 → 27.04). The values closest to $x = 5$ are very close to 25. Choice B confuses the input with the output value. Choices C and D incorrectly select specific table values rather than recognizing the limiting value that both sides approach.

Question 2

The following table gives values of $s(x)$ as $x$ approaches 4:

$x$	3.5	3.9	3.99	4.01	4.1	4.5
$s(x)$	-1.5	-1.9	-1.99	-1.99	-1.9	-1.5

Based on this information, what is $\lim_{x \to 4} s(x)$ ?

$-2$ since the values approach this from both directions (correct answer)
$-1.5$ based on the symmetric boundary values
$4$ because that's the input value we're approaching
$-1.99$ since it appears twice in the table

Explanation: The function values approach -2 as $x$ approaches 4. From both sides, the pattern shows: -1.5 → -1.9 → -1.99 (approaching -2). The symmetry in the table confirms this convergence pattern. Choice B incorrectly focuses on the outer values rather than the limiting behavior. Choice C confuses input with output. Choice D selects a table value rather than recognizing the limit that the values approach.

Question 3

Function $v(x)$ has these values near $x = -3$ :

$x$	-3.2	-3.1	-3.01	-2.99	-2.9	-2.8
$v(x)$	10.24	9.61	9.0601	8.9401	8.41	7.84

The limit $\lim_{x \to -3} v(x)$ is best estimated as:

$-3$ since that's the approach point for $x$
$9$ based on the convergence of the function values (correct answer)
$9.61$ which is the closest left-hand table value
$8.41$ which is the closest right-hand table value

Explanation: The values converge to 9 as $x$ approaches -3. From the left: 10.24 → 9.61 → 9.0601, and from the right: 8.9401 → 8.41 → 7.84. The values closest to $x = -3$ are 9.0601 and 8.9401, both near 9. Choice A confuses input with output. Choices C and D select specific table values from one side rather than identifying where both sides converge.

Question 4

The table below shows values of $f(x)$ near $x = 1.5$ :

$x$	1.2	1.4	1.49	1.51	1.6	1.8
$f(x)$	1.44	1.96	2.2201	2.2801	2.56	3.24

What is $\lim_{x \to 1.5} f(x)$ ?

$1.5$ because that matches the input approach value
$2.25$ since both sides converge to this value (correct answer)
$2.2201$ which is the nearest left approach value
$2.2801$ which is the nearest right approach value

Explanation: The function values approach 2.25 as $x$ approaches 1.5 from both directions. From the left: 1.44 → 1.96 → 2.2201, and from the right: 2.2801 → 2.56 → 3.24. The values 2.2201 and 2.2801 are both very close to 2.25. Choice A confuses the input with the output. Choices C and D select specific table values rather than the common limit value both sides approach.

Question 5

The table shows values of $d(x)$ as $x$ approaches 0.2:

$x$	0.1	0.19	0.199	0.201	0.21	0.3
$d(x)$	0.01	0.0361	0.039601	0.040401	0.0441	0.09

From this data, $\lim_{x \to 0.2} d(x)$ can be estimated as:

$0.2$ since that's the convergence point for $x$
$0.04$ based on the limiting behavior shown (correct answer)
$0.039601$ which gives the left-side precision
$0.040401$ which gives the right-side precision

Explanation: The function values approach 0.04 as $x$ approaches 0.2 from both sides. From the left: 0.01 → 0.0361 → 0.039601, and from the right: 0.040401 → 0.0441 → 0.09. The values closest to $x = 0.2$ are 0.039601 and 0.040401, both very close to 0.04. Choice A confuses the input value with the limit. Choices C and D select specific table values rather than the common limit value.

Question 6

Function $t(x)$ has the following values approaching $x = -0.5$ :

$x$	-0.7	-0.6	-0.51	-0.49	-0.4	-0.3
$t(x)$	0.49	0.36	0.2601	0.2401	0.16	0.09

What is $\lim_{x \to -0.5} t(x)$ ?

$-0.5$ because that's the input approach value
$0.36$ based on the closest left table entry
$0.25$ since the values converge to this number (correct answer)
$0.16$ based on the closest right table entry

Explanation: The function values approach 0.25 as $x$ approaches -0.5 from both directions. From the left: 0.49 → 0.36 → 0.2601, and from the right: 0.2401 → 0.16 → 0.09. The values nearest to $x = -0.5$ are 0.2601 and 0.2401, both very close to 0.25. Choice A confuses input with output. Choices B and D select specific table values from one side rather than the common limiting value.

Question 7

Function $b(x)$ has the following values near $x = 0.8$ :

$x$	0.6	0.7	0.79	0.81	0.9	1.0
$b(x)$	0.36	0.49	0.6241	0.6561	0.81	1.0

What is $\lim_{x \to 0.8} b(x)$ ?

$0.8$ since that matches the input approach value
$0.6241$ which is the most precise left value
$0.64$ since the values converge to this number (correct answer)
$0.6561$ which is the most precise right value

Explanation: The function values approach 0.64 as $x$ approaches 0.8 from both sides. From the left: 0.36 → 0.49 → 0.6241, and from the right: 0.6561 → 0.81 → 1.0. The values nearest to $x = 0.8$ are 0.6241 and 0.6561, both very close to 0.64. Choice A confuses input with output values. Choices B and D select specific table entries rather than identifying the common limit value.

Question 8

Function $z(x)$ has these values near $x = -1.5$ :

$x$	-1.8	-1.6	-1.51	-1.49	-1.4	-1.2
$z(x)$	3.24	2.56	2.2801	2.2201	1.96	1.44

Based on this data, what is $\lim_{x \to -1.5} z(x)$ ?

$-1.5$ since that's where $x$ is approaching
$2.25$ as indicated by the convergence pattern (correct answer)
$2.2801$ which is the closest left table value
$2.2201$ which is the closest right table value

Explanation: The values approach 2.25 as $x$ approaches -1.5 from both sides. From the left: 3.24 → 2.56 → 2.2801, and from the right: 2.2201 → 1.96 → 1.44. The values nearest $x = -1.5$ are 2.2801 and 2.2201, both very close to 2.25. Choice A confuses input with output. Choices C and D select specific table values rather than recognizing the common limit value both sides approach.

Question 9

The following table shows values of $a(x)$ approaching $x = 3.2$ :

$x$	3.0	3.1	3.19	3.21	3.3	3.4
$a(x)$	9.0	9.61	10.1761	10.2241	10.89	11.56

The limit $\lim_{x \to 3.2} a(x)$ can be estimated as:

$3.2$ because that's the point of approach
$9.61$ from the nearest left computational value
$10.2$ based on the convergence behavior shown (correct answer)
$10.89$ from the nearest right computational value

Explanation: The function values approach 10.2 as $x$ approaches 3.2 from both directions. From the left: 9.0 → 9.61 → 10.1761, and from the right: 10.2241 → 10.89 → 11.56. The values closest to $x = 3.2$ are 10.1761 and 10.2241, both very near 10.2. Choice A confuses input with output. Choices B and D select specific table values from one side rather than the common limiting value.

Question 10

Consider function $q(x)$ with values near $x = 6$ :

$x$	5.7	5.9	5.99	6.01	6.1	6.3
$q(x)$	32.49	34.81	35.8801	36.1201	37.21	39.69

Based on the table, $\lim_{x \to 6} q(x)$ equals:

$6$ since that's where the function is evaluated
$35.8801$ from the most accurate left computation
$36$ as indicated by the convergence pattern (correct answer)
$36.1201$ from the most accurate right computation

Explanation: The values show convergence to 36 as $x$ approaches 6. From the left: 32.49 → 34.81 → 35.8801, and from the right: 36.1201 → 37.21 → 39.69. The values nearest $x = 6$ are 35.8801 and 36.1201, both very close to 36. Choice A confuses input with output. Choices B and D select specific table entries rather than recognizing the value that both sides approach.

Question 11

A function $g(x)$ has the following values near $x = -2$ :

$x$	-2.1	-2.01	-2.001	-1.999	-1.99	-1.9
$g(x)$	4.21	4.0201	4.002001	3.997999	3.9801	3.81

What is $\lim_{x \to -2} g(x)$ ?

$-2$ since that's where the function is centered
$4$ based on the convergence pattern shown (correct answer)
$4.21$ which is the leftmost value in the table
The limit is undefined due to jump discontinuity

Explanation: The table shows that as $x$ approaches -2 from both directions, $g(x)$ approaches 4. From the left: 4.21 → 4.0201 → 4.002001, and from the right: 3.997999 → 3.9801 → 3.81. Both one-sided limits approach 4. Choice A confuses the input with the output. Choice C incorrectly selects a specific table value rather than the limit. Choice D is wrong because the values show smooth convergence to the same value from both sides.

Question 12

The table below shows values of a function $f(x)$ near $x = 3$ .

$x$	2.9	2.99	2.999	3.001	3.01	3.1
$f(x)$	7.41	7.9401	7.994001	8.006001	8.0601	8.61

Based on the table, what is the best estimate for $\lim_{x \to 3} f(x)$ ?

$8$ (correct answer)
$7.5$ because it's the average of the boundary values
$3$ since that's the value we're approaching
The limit does not exist due to oscillating behavior

Explanation: As $x$ approaches 3 from both sides, the function values approach 8. From the left: 7.41 → 7.9401 → 7.994001, and from the right: 8.006001 → 8.0601 → 8.61. The values are getting closer to 8 as $x$ gets closer to 3. Choice B incorrectly averages boundary values rather than examining the limit behavior. Choice C confuses the input value with the limit value. Choice D is incorrect because the values show clear convergence, not oscillation.

Question 13

Values of function $y(x)$ as $x$ approaches 2.5 are shown:

$x$	2.3	2.4	2.49	2.51	2.6	2.7
$y(x)$	5.29	5.76	6.2001	6.3001	6.76	7.29

From this table, $\lim_{x \to 2.5} y(x)$ is:

$2.5$ because that's the input convergence point
$6.2001$ which is the precise left approach value
$6.25$ since both sides converge toward this value (correct answer)
$6.3001$ which is the precise right approach value

Explanation: The function values approach 6.25 as $x$ approaches 2.5 from both directions. From the left: 5.29 → 5.76 → 6.2001, and from the right: 6.3001 → 6.76 → 7.29. The values nearest to $x = 2.5$ are 6.2001 and 6.3001, both very close to 6.25. Choice A confuses input with output values. Choices B and D select specific table entries rather than the value both sides approach.

Question 14

Values of function $w(x)$ near $x = 0.5$ are shown below:

$x$	0.4	0.49	0.499	0.501	0.51	0.6
$w(x)$	1.6	1.96	1.996	2.004	2.04	2.4

From this table, $\lim_{x \to 0.5} w(x)$ can be estimated as:

$1.996$ which is the most precise left value
$2$ based on the pattern of convergence shown (correct answer)
$2.004$ which is the most precise right value
$0.5$ since that's the point of convergence

Explanation: The table shows convergence to 2 from both sides. As $x$ approaches 0.5 from the left: 1.6 → 1.96 → 1.996, and from the right: 2.004 → 2.04 → 2.4. The values closest to $x = 0.5$ are 1.996 and 2.004, both very near 2. Choices A and C select specific table entries rather than the limiting value. Choice D confuses the input value with the function's limit value.

Question 15

Function $k(x)$ has the following values approaching $x = -1$ :

$x$	-1.3	-1.1	-1.01	-0.99	-0.9	-0.7
$k(x)$	8.69	9.21	9.9801	10.0201	10.81	12.49

Based on this data, $\lim_{x \to -1} k(x)$ equals:

$9.21$ since it's from the nearest left approach
$10$ as indicated by the convergence behavior (correct answer)
$10.81$ since it's from the nearest right approach
The limit cannot be determined from this sparse data

Explanation: The values show clear convergence to 10 as $x$ approaches -1. From the left: 8.69 → 9.21 → 9.9801, and from the right: 10.0201 → 10.81 → 12.49. The values closest to $x = -1$ are 9.9801 and 10.0201, which both approach 10. Choices A and C incorrectly select specific table values rather than the limit. Choice D is wrong because the data clearly shows convergent behavior toward 10.

Question 16

The table shows values of $h(x)$ as $x$ approaches 1:

$x$	0.9	0.99	0.999	1.001	1.01	1.1
$h(x)$	-0.526	-0.503	-0.500	0.500	0.503	0.526

Based on this information, $\lim_{x \to 1} h(x)$ is:

$0$ since the values are symmetric around zero
$-0.5$ based on the left-hand approach pattern
$0.5$ following the right-hand approach trend
Does not exist because left and right limits differ (correct answer)

Explanation: The left-hand limit is $\lim_{x \to 1^-} h(x) = -0.5$ (values approach -0.5 from the left), while the right-hand limit is $\lim_{x \to 1^+} h(x) = 0.5$ (values approach 0.5 from the right). Since these one-sided limits are different, the two-sided limit does not exist. Choice A incorrectly assumes symmetry implies a limit of zero. Choices B and C only consider one-sided behavior and ignore the requirement that both sides must agree for the limit to exist.

Question 17

The table shows values of $m(x)$ near $x = 2$ :

$x$	1.7	1.9	1.99	2.01	2.1	2.3
$m(x)$	2.89	3.61	3.9601	4.0401	4.41	5.29

What is the most reasonable estimate for $\lim_{x \to 2} m(x)$ ?

$2$ because that's the value of $x$ we approach
$3.61$ based on the left-hand side approximation
$4$ since both sides converge toward this value (correct answer)
$4.41$ based on the right-hand side approximation

Explanation: The function values approach 4 as $x$ approaches 2 from both directions. From the left: 2.89 → 3.61 → 3.9601, and from the right: 4.0401 → 4.41 → 5.29. The values nearest to $x = 2$ are 3.9601 and 4.0401, both very close to 4. Choice A confuses input and output values. Choices B and D select specific table values from one side rather than identifying the common limit value.

Question 18

The table below gives values of $u(x)$ near $x = 7$ :

$x$	6.5	6.9	6.99	7.01	7.1	7.5
$u(x)$	42.25	47.61	48.8601	49.1401	50.41	56.25

The best estimate for $\lim_{x \to 7} u(x)$ is:

$7$ since that's the point we're approaching
$47.61$ from the nearest left-hand calculation
$49$ based on the convergence shown in the table (correct answer)
$50.41$ from the nearest right-hand calculation

Explanation: The values converge to 49 as $x$ approaches 7 from both sides. From the left: 42.25 → 47.61 → 48.8601, and from the right: 49.1401 → 50.41 → 56.25. The values closest to $x = 7$ are 48.8601 and 49.1401, both very close to 49. Choice A confuses input with output. Choices B and D select specific table values from one side rather than identifying the common limit value.

Question 19

Consider the following table of values for function $p(x)$ near $x = 0$ :

$x$	-0.1	-0.01	-0.001	0.001	0.01	0.1
$p(x)$	0.9950	0.9999	1.0000	1.0000	0.9999	0.9950

What can be concluded about $\lim_{x \to 0} p(x)$ ?

The limit equals $0.995$ based on the outer values
The limit equals $1$ since values converge to this number (correct answer)
The limit equals $0$ since we're approaching $x = 0$
The limit is undefined due to insufficient precision shown

Explanation: As $x$ approaches 0 from both sides, the function values approach 1. The pattern shows 0.9950 → 0.9999 → 1.0000 from the left, and 1.0000 → 0.9999 → 0.9950 from the right, indicating convergence to 1. Choice A incorrectly focuses on the boundary values rather than the limiting behavior. Choice C confuses the input value with the limit value. Choice D is incorrect because the precision shown is sufficient to determine the limit approaches 1.