Estimating Limit Values from Tables - AP Calculus AB

1. 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:
   Answer: $2$ based on the pattern of convergence shown 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.
2. 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 $\lim_{x \to 3} f(x)$?
   Answer: $8$ 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.
3. 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:
   Answer: Does not exist because left and right limits differ 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.
4. 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 $\lim_{x \to -2} g(x)$?
   Answer: $4$ based on the convergence pattern shown 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.
5. 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:
   Answer: $6.25$ since both sides converge toward this value 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.
6. 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 $\lim_{x \to 0} p(x)$?
   Answer: The limit equals $1$ since values converge to this number 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.
7. 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 $\lim_{x \to -0.5} t(x)$?
   Answer: $0.25$ since the values converge to this number 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.
8. 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:
   Answer: $10$ as indicated by the convergence behavior 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.
9. 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 $\lim_{x \to -1.5} z(x)$?
   Answer: $2.25$ as indicated by the convergence pattern 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.
10. 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:
    Answer: $0.04$ based on the limiting behavior shown 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.
11. 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:
    Answer: $10.2$ based on the convergence behavior shown 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.
12. 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:
    Answer: $25$ based on the convergence pattern observed 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.
13. 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:
    Answer: $49$ based on the convergence shown in the table 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.
14. 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 $\lim_{x \to 4} s(x)$?
    Answer: $-2$ since the values approach this from both directions 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.
15. 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 $\lim_{x \to 1.5} f(x)$?
    Answer: $2.25$ since both sides converge to this value 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.
16. 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:
    Answer: $36$ as indicated by the convergence pattern 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.
17. 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 $\lim_{x \to 0.8} b(x)$?
    Answer: $0.64$ since the values converge to this number 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.
18. 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 $\lim_{x \to 2} m(x)$?
    Answer: $4$ since both sides converge toward this value 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.
19. 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:
    Answer: $9$ based on the convergence of the function values 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.

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