AP Calculus BC Quiz: Estimating Limit Values From Tables
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Estimating Limit Values From TablesQuestion 1 of 15

Values of a function gg are given for selected values of xx near -4. For xx values of -4.1, -4.01, and -4.001, the g(x)g(x) values are 500, 5000, and 50000. For xx values of -3.999, -3.99, and -3.9, the g(x)g(x) values are 49999, 4999, and 499. The function is undefined at x=−4x=-4. What is the best estimate for lim⁡x→−4g(x)\lim_{x \to -4} g(x)?

50000
∞\infty
0
The limit does not exist.
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AP Calculus BC Quiz

AP Calculus BC Quiz: Estimating Limit Values From Tables

Practice Estimating Limit Values From Tables 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 Tables, 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

Values of a function gg are given for selected values of xx near -4. For xx values of -4.1, -4.01, and -4.001, the g(x)g(x) values are 500, 5000, and 50000. For xx values of -3.999, -3.99, and -3.9, the g(x)g(x) values are 49999, 4999, and 499. The function is undefined at x=−4x=-4. What is the best estimate for lim⁡x→−4g(x)\lim_{x \to -4} g(x)?

  1. 50000
  2. ∞\infty (correct answer)
  3. 0
  4. The limit does not exist.
Explanation: The correct answer is ∞\infty. As xx approaches -4 from the left, the values of g(x)g(x) are increasing without bound. As xx approaches -4 from the right, the values of g(x)g(x) are also increasing without bound. Since both sides approach positive infinity, the limit is considered to be ∞\infty. Choice A is one of the data points. Choice D is incorrect because the behavior from both sides is the same (approaching ∞\infty).

Question 2

The function hh is continuous. A table of values for h(x)h(x) is given: for xx values of -10, -100, -1000, and -10000, the corresponding h(x)h(x) values are -0.4, -0.49, -0.499, and -0.4999. What is the best estimate for lim⁡x→−∞h(x)\lim_{x \to -\infty} h(x)?

  1. -0.5 (correct answer)
  2. 0
  3. -0.4
  4. The limit does not exist.
Explanation: The correct answer is -0.5. The limit as x→−∞x \to -\infty describes the end behavior of the function as xx decreases without bound. The table shows that as xx becomes more negative, the values of h(x)h(x) get closer to -0.5. Choice C is the first value in the table. Choice B is a common limit value but not supported by the data. Choice D is incorrect as the values are approaching a single number.

Question 3

The functions ff and gg are continuous. A table of values near x=−3x=-3 is given. For xx values of -3.1, -3.01, and -3.001, the f(x)f(x) values are 11.7, 11.97, and 11.997, and the g(x)g(x) values are 1.9, 1.99, and 1.999. For xx values of -2.999, -2.99, and -2.9, the f(x)f(x) values are 12.003, 12.03, and 12.3, and the g(x)g(x) values are 2.001, 2.01, and 2.1. What is the best estimate for lim⁡x→−3f(x)g(x)\lim_{x \to -3} \frac{f(x)}{g(x)}?

  1. 6 (correct answer)
  2. 10
  3. 14
  4. The limit does not exist.
Explanation: The correct answer is 6. From the table, as x→−3x \to -3, f(x)→12f(x) \to 12 and g(x)→2g(x) \to 2. Using the quotient property of limits, lim⁡x→−3f(x)g(x)=lim⁡x→−3f(x)lim⁡x→−3g(x)=122=6\lim_{x \to -3} \frac{f(x)}{g(x)} = \frac{\lim_{x \to -3} f(x)}{\lim_{x \to -3} g(x)} = \frac{12}{2} = 6. Choice B is the sum of the limits. Choice C is the sum of the function values at x=-3 (if they were given). Choice D is incorrect because the individual limits exist and the denominator's limit is not zero.

Question 4

Values of a function hh are given for selected values of xx near 3. For xx values of 2.9, 2.99, and 2.999, the corresponding h(x)h(x) values are -1.5, -1.95, and -1.995. For xx values of 3.001, 3.01, and 3.1, the corresponding h(x)h(x) values are -3.998, -3.98, and -3.8. The function is undefined at x=3x=3. What is the best estimate for lim⁡x→3+h(x)\lim_{x \to 3^+} h(x)?

  1. -2
  2. -4 (correct answer)
  3. 3
  4. The limit does not exist.
Explanation: The correct answer is -4. The notation x→3+x \to 3^+ indicates the limit as xx approaches 3 from the right side (values greater than 3). The table shows that for xx values of 3.001, 3.01, and 3.1, the values of h(x)h(x) are approaching -4. Choice A is the left-hand limit. Choice C is the value xx is approaching. Choice D is incorrect because the one-sided limit exists.

Question 5

A differentiable function ff satisfies f(2)=8f(2) = 8. A table of values for the difference quotient f(2+h)−f(2)h\frac{f(2+h)-f(2)}{h} is given. For hh values of -0.1, -0.01, and -0.001, the quotient's values are 3.9, 3.99, and 3.999. For hh values of 0.001, 0.01, and 0.1, the quotient's values are 4.001, 4.01, and 4.1. What is the best estimate for lim⁡h→0f(2+h)−f(2)h\lim_{h \to 0} \frac{f(2+h)-f(2)}{h}?

  1. 0
  2. 4 (correct answer)
  3. 8
  4. The limit does not exist.
Explanation: The correct answer is 4. The expression in the limit is the definition of the derivative of ff at x=2x=2, i.e., f′(2)f'(2). The table directly provides values of this difference quotient as hh approaches 0 from the left and the right. As h→0−h \to 0^-, the quotient approaches 4. As h→0+h \to 0^+, the quotient approaches 4. Since both one-sided limits are equal, the limit is 4. Choice C is the value of f(2)f(2).

Question 6

The functions ff and gg are continuous. A table of values near x=4x=4 is given. For xx values of 3.9, 3.99, and 3.999, the f(x)f(x) values are 6.8, 6.98, and 6.998, and the g(x)g(x) values are -2.2, -2.02, and -2.002. For xx values of 4.001, 4.01, and 4.1, the f(x)f(x) values are 7.002, 7.02, and 7.2, and the g(x)g(x) values are -1.998, -1.98, and -1.8. What is the best estimate for lim⁡x→4(f(x)+g(x))\lim_{x \to 4} (f(x) + g(x))?

  1. 5 (correct answer)
  2. 9
  3. 4.6
  4. The limit does not exist.
Explanation: The correct answer is 5. From the table, as x→4x \to 4, f(x)→7f(x) \to 7 and g(x)→−2g(x) \to -2. Using the sum property of limits, lim⁡x→4(f(x)+g(x))=lim⁡x→4f(x)+lim⁡x→4g(x)=7+(−2)=5\lim_{x \to 4} (f(x) + g(x)) = \lim_{x \to 4} f(x) + \lim_{x \to 4} g(x) = 7 + (-2) = 5. Choice B is the difference of the limits. Choice C is the sum of the first values in the table. Choice D is incorrect because both individual limits exist.

Question 7

Values of a function ff are given for selected values of xx near 2. The table shows that for xx values of 1.9, 1.99, and 1.999, the corresponding f(x)f(x) values are 4.71, 4.97, and 4.997. For xx values of 2.001, 2.01, and 2.1, the corresponding f(x)f(x) values are 5.003, 5.03, and 5.31. The value of f(2)f(2) is 7. What is the best estimate for lim⁡x→2f(x)\lim_{x \to 2} f(x)?

  1. 5 (correct answer)
  2. 7
  3. 5.003
  4. The limit does not exist.
Explanation: The correct answer is 5. As xx approaches 2 from the left (x=1.9,1.99,1.999x=1.9, 1.99, 1.999), f(x)f(x) approaches 5. As xx approaches 2 from the right (x=2.001,2.01,2.1x=2.001, 2.01, 2.1), f(x)f(x) also approaches 5. Since the left-hand and right-hand limits are equal, the limit is 5. The value f(2)=7f(2)=7 is irrelevant to the value of the limit. Choice B is the value of the function at x=2x=2, not the limit. Choice C is a single value from the table, not the limit. Choice D is incorrect because the left and right limits both approach 5.

Question 8

Values of a function gg are given for selected values of xx near -1. For xx values of -1.1, -1.01, and -1.001, the corresponding g(x)g(x) values are 8.8, 8.98, and 8.998. For xx values of -0.999, -0.99, and -0.9, the corresponding g(x)g(x) values are 7.002, 7.02, and 7.2. The value of g(−1)g(-1) is 3. What is the best estimate for lim⁡x→−1−g(x)\lim_{x \to -1^-} g(x)?

  1. 3
  2. 7
  3. 9 (correct answer)
  4. The limit does not exist.
Explanation: The correct answer is 9. The notation x→−1−x \to -1^- indicates the limit as xx approaches -1 from the left side (values less than -1). The table shows that for xx values of -1.1, -1.01, and -1.001, the values of g(x)g(x) are approaching 9. Choice A is the value of the function at x=−1x=-1. Choice B is the right-hand limit, not the left-hand limit. Choice D is incorrect because the one-sided limit exists.

Question 9

The function ff is continuous. A table of values for f(x)f(x) is given: for xx values of 10, 100, 1000, and 10000, the corresponding f(x)f(x) values are 3.1, 3.01, 3.001, and 3.0001. What is the best estimate for lim⁡x→∞f(x)\lim_{x \to \infty} f(x)?

  1. 0
  2. 3 (correct answer)
  3. 3.1
  4. The limit does not exist.
Explanation: The correct answer is 3. The limit as x→∞x \to \infty describes the end behavior of the function as xx increases without bound. The table shows that as xx gets larger, the values of f(x)f(x) get closer and closer to 3. Choice A is a common limit value but not supported by the data. Choice C is the first value in the table. Choice D is incorrect as the values are approaching a single number.

Question 10

The functions ff and gg are continuous. A table of values near x=4x=4 is given. For xx values of 3.9, 3.99, and 3.999, the f(x)f(x) values are 6.8, 6.98, and 6.998, and the g(x)g(x) values are -2.2, -2.02, and -2.002. For xx values of 4.001, 4.01, and 4.1, the f(x)f(x) values are 7.002, 7.02, and 7.2, and the g(x)g(x) values are -1.998, -1.98, and -1.8. What is the best estimate for lim⁡x→4(f(x)⋅g(x))\lim_{x \to 4} (f(x) \cdot g(x))?

  1. -14 (correct answer)
  2. 5
  3. -14.96
  4. The limit does not exist.
Explanation: The correct answer is -14. From the table, as x→4x \to 4, f(x)→7f(x) \to 7 and g(x)→−2g(x) \to -2. Using the product property of limits, lim⁡x→4(f(x)⋅g(x))=(lim⁡x→4f(x))⋅(lim⁡x→4g(x))=7⋅(−2)=−14\lim_{x \to 4} (f(x) \cdot g(x)) = (\lim_{x \to 4} f(x)) \cdot (\lim_{x \to 4} g(x)) = 7 \cdot (-2) = -14. Choice B is the sum of the limits. Choice C is the product of the first values in the table. Choice D is incorrect because both individual limits exist.

Question 11

The functions ff and gg are continuous. For xx values of 1.9, 1.99, and 1.999, the g(x)g(x) values are 2.8, 2.98, and 2.998. For xx values of 2.001, 2.01, and 2.1, the g(x)g(x) values are 3.002, 3.02, and 3.2. For yy values of 2.9, 2.99, and 2.999, the f(y)f(y) values are 8.7, 8.97, and 8.997. For yy values of 3.001, 3.01, and 3.1, the f(y)f(y) values are 9.003, 9.03, and 9.3. What is the best estimate for lim⁡x→2f(g(x))\lim_{x \to 2} f(g(x))?

  1. 2
  2. 3
  3. 9 (correct answer)
  4. The limit does not exist.
Explanation: The correct answer is 9. To evaluate the limit of a composite function, first evaluate the inner limit: lim⁡x→2g(x)\lim_{x \to 2} g(x). From the table for g(x)g(x), as x→2x \to 2, g(x)→3g(x) \to 3. Now, evaluate the outer limit using this result: lim⁡y→3f(y)\lim_{y \to 3} f(y). From the table for f(y)f(y), as y→3y \to 3, f(y)→9f(y) \to 9. Therefore, lim⁡x→2f(g(x))=9\lim_{x \to 2} f(g(x)) = 9. Choice B is the limit of the inner function. Choice A is the value x is approaching.

Question 12

The function ff is continuous. A table of values for f(x)f(x) near x=0x=0 is given. For xx values of -0.1, -0.01, and -0.001, the f(x)f(x) values are 4.8, 4.98, and 4.998. For xx values of 0.001, 0.01, and 0.1, the f(x)f(x) values are 5.002, 5.02, and 5.2. What is the best estimate for lim⁡x→0(ex⋅f(x))\lim_{x \to 0} (e^x \cdot f(x))?

  1. 0
  2. 5 (correct answer)
  3. 1
  4. The limit does not exist.
Explanation: The correct answer is 5. From the table, we estimate that lim⁡x→0f(x)=5\lim_{x \to 0} f(x) = 5. For the known function exe^x, we know that lim⁡x→0ex=e0=1\lim_{x \to 0} e^x = e^0 = 1. By the product property of limits, lim⁡x→0(ex⋅f(x))=(lim⁡x→0ex)⋅(lim⁡x→0f(x))=1⋅5=5\lim_{x \to 0} (e^x \cdot f(x)) = (\lim_{x \to 0} e^x) \cdot (\lim_{x \to 0} f(x)) = 1 \cdot 5 = 5. Choice C is the limit of exe^x. Choice A would be true if one of the limits were 0. Choice D is incorrect because both individual limits exist.

Question 13

The function gg has a removable discontinuity at x=1x=1. For xx values of 0.9, 0.99, and 0.999, the g(x)g(x) values are -2.19, -2.0199, and -2.001999. For xx values of 1.001, 1.01, and 1.1, the g(x)g(x) values are -1.997999, -1.9799, and -1.79. The value of g(1)g(1) is 5. What is the best estimate for lim⁡x→1g(x)\lim_{x \to 1} g(x)?

  1. 5
  2. -1.79
  3. -2 (correct answer)
  4. The limit does not exist.
Explanation: The correct answer is -2. As xx approaches 1 from the left, the values of g(x)g(x) approach -2. As xx approaches 1 from the right, the values of g(x)g(x) also approach -2. Since both one-sided limits are equal to -2, the limit is -2. The fact that g(1)=5g(1)=5 creates a 'hole' in the graph but does not affect the limit value. Choice A is the function value. Choice B is one of the table values.

Question 14

The functions ff and gg are continuous. For xx values of 0.9, 0.99, and 0.999, the f(x)f(x) values are 3.8, 3.98, and 3.998. For xx values of 1.001, 1.01, and 1.1, the f(x)f(x) values are 4.002, 4.02, and 4.2. For yy values of 3.9, 3.99, and 3.999, the g(y)g(y) values are -5.2, -5.02, and -5.002. For yy values of 4.001, 4.01, and 4.1, the g(y)g(y) values are -4.998, -4.98, and -4.8. What is the best estimate for lim⁡x→1g(f(x))\lim_{x \to 1} g(f(x))?

  1. -5 (correct answer)
  2. 4
  3. 1
  4. The limit does not exist.
Explanation: The correct answer is -5. First, we find the limit of the inner function, lim⁡x→1f(x)\lim_{x \to 1} f(x). The table for f(x)f(x) shows that as xx approaches 1, f(x)f(x) approaches 4. Next, we use this value as the input for the outer function's limit: lim⁡y→4g(y)\lim_{y \to 4} g(y). The table for g(y)g(y) shows that as yy approaches 4, g(y)g(y) approaches -5. Thus, lim⁡x→1g(f(x))=−5\lim_{x \to 1} g(f(x)) = -5. Choice B is the limit of the inner function.

Question 15

A function ff is continuous. For xx values of 2.9, 2.99, and 2.999, the values for f(x)f(x) are -0.1, -0.01, and -0.001. For xx values of 3.001, 3.01, and 3.1, the values for f(x)f(x) are 0.001, 0.01, and 0.1. We know f(3)=0f(3)=0. What is the best estimate for lim⁡x→3sin⁡(f(x))f(x)\lim_{x \to 3} \frac{\sin(f(x))}{f(x)}?

  1. 0
  2. 1 (correct answer)
  3. 3
  4. The limit does not exist.
Explanation: The correct answer is 1. We can use a change of variable. Let u=f(x)u = f(x). From the table, as x→3x \to 3, the values of f(x)f(x) approach 0. So, as x→3x \to 3, we have u→0u \to 0. The limit can be rewritten as lim⁡u→0sin⁡(u)u\lim_{u \to 0} \frac{\sin(u)}{u}. This is a fundamental trigonometric limit which equals 1. Choice A is the limit of the function f(x)f(x). Choice C is the value xx approaches.