Estimating Limit Values from Tables
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AP Calculus AB › Estimating Limit Values from Tables
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:
$$24.9801$$ which is the closest left-side value
$$25$$ based on the convergence pattern observed
$$5$$ since that's the value $$x$$ approaches
$$25.0201$$ 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.
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 $$\lim_{x \to 4} s(x)$$?
$$-1.99$$ since it appears twice in the table
$$-2$$ since the values approach this from both directions
$$-1.5$$ based on the symmetric boundary values
$$4$$ because that's the input value we're approaching
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.
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:
$$8.41$$ which is the closest right-hand table value
$$9$$ based on the convergence of the function values
$$9.61$$ which is the closest left-hand table value
$$-3$$ since that's the approach point for $$x$$
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.
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 $$\lim_{x \to 1.5} f(x)$$?
$$2.25$$ since both sides converge to this value
$$2.2201$$ which is the nearest left approach value
$$1.5$$ because that matches the input 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.
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 $$x$$
$$0.040401$$ which gives the right-side precision
$$0.04$$ based on the limiting behavior shown
$$0.039601$$ which gives the left-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.
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 $$\lim_{x \to -0.5} t(x)$$?
$$0.16$$ based on the closest right table entry
$$-0.5$$ because that's the input approach value
$$0.25$$ since the values converge to this number
$$0.36$$ based on the closest left 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.
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 $$\lim_{x \to 0.8} b(x)$$?
$$0.64$$ since the values converge to this number
$$0.8$$ since that matches the input approach value
$$0.6561$$ which is the most precise right value
$$0.6241$$ which is the most precise left 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.
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 $$\lim_{x \to -1.5} z(x)$$?
$$2.25$$ as indicated by the convergence pattern
$$2.2801$$ which is the closest left table value
$$2.2201$$ which is the closest right table value
$$-1.5$$ since that's where $$x$$ is approaching
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.
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:
$$10.89$$ from the nearest right computational value
$$9.61$$ from the nearest left computational value
$$10.2$$ based on the convergence behavior shown
$$3.2$$ because that's the point of approach
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.
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:
$$35.8801$$ from the most accurate left computation
$$36$$ as indicated by the convergence pattern
$$6$$ since that's where the function is evaluated
$$36.1201$$ 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.