AP Calculus BC Quiz: Estimating Limit Values From Tables

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.

Question 1

1. 50000
2. $\infty$ (correct answer)
3. 0
4. The limit does not exist.

Explanation: The correct answer is $\infty$. As $x$ approaches -4 from the left, the values of $g(x)$ are increasing without bound. As $x$ approaches -4 from the right, the values of $g(x)$ 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

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 $x \to -\infty$ describes the end behavior of the function as $x$ decreases without bound. The table shows that as $x$ becomes more negative, the values of $h(x)$ 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

1. 6 (correct answer)
2. 10
3. 14
4. The limit does not exist.

Explanation: The correct answer is 6. From the table, as $x \to -3$, $f(x) \to 12$ and $g(x) \to 2$. Using the quotient property of limits, $\lim_{x \to -3} \frac{f(x)}{g(x)} = \frac{\lim_{x \to -3} f(x)}{\lim_{x \to -3} g(x)} = \frac{12}{2} = 6$. 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

1. -2
2. -4 (correct answer)
3. 3
4. The limit does not exist.

Explanation: The correct answer is -4. The notation $x \to 3^+$ indicates the limit as $x$ approaches 3 from the right side (values greater than 3). The table shows that for $x$ values of 3.001, 3.01, and 3.1, the values of $h(x)$ are approaching -4. Choice A is the left-hand limit. Choice C is the value $x$ is approaching. Choice D is incorrect because the one-sided limit exists.

Question 5

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 $f$ at $x=2$, i.e., $f'(2)$. The table directly provides values of this difference quotient as $h$ approaches 0 from the left and the right. As $h \to 0^-$, the quotient approaches 4. As $h \to 0^+$, the quotient approaches 4. Since both one-sided limits are equal, the limit is 4. Choice C is the value of $f(2)$.

Question 6

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 $x \to 4$, $f(x) \to 7$ and $g(x) \to -2$. Using the sum property of limits, $\lim_{x \to 4} (f(x) + g(x)) = \lim_{x \to 4} f(x) + \lim_{x \to 4} g(x) = 7 + (-2) = 5$. 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

1. 5 (correct answer)
2. 7
3. 5.003
4. The limit does not exist.

Explanation: The correct answer is 5. As $x$ approaches 2 from the left ($x=1.9, 1.99, 1.999$), $f(x)$ approaches 5. As $x$ approaches 2 from the right ($x=2.001, 2.01, 2.1$), $f(x)$ also approaches 5. Since the left-hand and right-hand limits are equal, the limit is 5. The value $f(2)=7$ is irrelevant to the value of the limit. Choice B is the value of the function at $x=2$, 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

1. 3
2. 7
3. 9 (correct answer)
4. The limit does not exist.

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

Question 9

1. 0
2. 3 (correct answer)
3. 3.1
4. The limit does not exist.

Explanation: The correct answer is 3. The limit as $x \to \infty$ describes the end behavior of the function as $x$ increases without bound. The table shows that as $x$ gets larger, the values of $f(x)$ 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

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 $x \to 4$, $f(x) \to 7$ and $g(x) \to -2$. Using the product property of limits, $\lim_{x \to 4} (f(x) \cdot g(x)) = (\lim_{x \to 4} f(x)) \cdot (\lim_{x \to 4} g(x)) = 7 \cdot (-2) = -14$. 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

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: $\lim_{x \to 2} g(x)$. From the table for $g(x)$, as $x \to 2$, $g(x) \to 3$. Now, evaluate the outer limit using this result: $\lim_{y \to 3} f(y)$. From the table for $f(y)$, as $y \to 3$, $f(y) \to 9$. Therefore, $\lim_{x \to 2} f(g(x)) = 9$. Choice B is the limit of the inner function. Choice A is the value x is approaching.

Question 12

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 $\lim_{x \to 0} f(x) = 5$. For the known function $e^x$, we know that $\lim_{x \to 0} e^x = e^0 = 1$. By the product property of limits, $\lim_{x \to 0} (e^x \cdot f(x)) = (\lim_{x \to 0} e^x) \cdot (\lim_{x \to 0} f(x)) = 1 \cdot 5 = 5$. Choice C is the limit of $e^x$. Choice A would be true if one of the limits were 0. Choice D is incorrect because both individual limits exist.

Question 13

1. 5
2. -1.79
3. -2 (correct answer)
4. The limit does not exist.

Explanation: The correct answer is -2. As $x$ approaches 1 from the left, the values of $g(x)$ approach -2. As $x$ approaches 1 from the right, the values of $g(x)$ also approach -2. Since both one-sided limits are equal to -2, the limit is -2. The fact that $g(1)=5$ 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

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, $\lim_{x \to 1} f(x)$. The table for $f(x)$ shows that as $x$ approaches 1, $f(x)$ approaches 4. Next, we use this value as the input for the outer function's limit: $\lim_{y \to 4} g(y)$. The table for $g(y)$ shows that as $y$ approaches 4, $g(y)$ approaches -5. Thus, $\lim_{x \to 1} g(f(x)) = -5$. Choice B is the limit of the inner function.

Question 15

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 $u = f(x)$. From the table, as $x \to 3$, the values of $f(x)$ approach 0. So, as $x \to 3$, we have $u \to 0$. The limit can be rewritten as $\lim_{u \to 0} \frac{\sin(u)}{u}$. This is a fundamental trigonometric limit which equals 1. Choice A is the limit of the function $f(x)$. Choice C is the value $x$ approaches.