Estimating Limit Values from Tables - AP Calculus BC

The function $h$ is continuous. A table of values for $h(x)$ is given: for $x$ values of -10, -100, -1000, and -10000, the corresponding $h(x)$ values are -0.4, -0.49, -0.499, and -0.4999. What is the best estimate for $\lim_{x \to -\infty} h(x)$ ?
Answer: -0.5 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.
Values of a function $g$ are given for selected values of $x$ near -1. For $x$ values of -1.1, -1.01, and -1.001, the corresponding $g(x)$ values are 8.8, 8.98, and 8.998. For $x$ values of -0.999, -0.99, and -0.9, the corresponding $g(x)$ values are 7.002, 7.02, and 7.2. The value of $g(-1)$ is 3. What is the best estimate for $\lim_{x \to -1^-} g(x)$ ?
Answer: 9 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.
The functions $f$ and $g$ are continuous. For $x$ values of 1.9, 1.99, and 1.999, the $g(x)$ values are 2.8, 2.98, and 2.998. For $x$ values of 2.001, 2.01, and 2.1, the $g(x)$ values are 3.002, 3.02, and 3.2. For $y$ values of 2.9, 2.99, and 2.999, the $f(y)$ values are 8.7, 8.97, and 8.997. For $y$ values of 3.001, 3.01, and 3.1, the $f(y)$ values are 9.003, 9.03, and 9.3. What is the best estimate for $\lim_{x \to 2} f(g(x))$ ?
Answer: 9 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.
The functions $f$ and $g$ are continuous. For $x$ values of 0.9, 0.99, and 0.999, the $f(x)$ values are 3.8, 3.98, and 3.998. For $x$ values of 1.001, 1.01, and 1.1, the $f(x)$ values are 4.002, 4.02, and 4.2. For $y$ values of 3.9, 3.99, and 3.999, the $g(y)$ values are -5.2, -5.02, and -5.002. For $y$ values of 4.001, 4.01, and 4.1, the $g(y)$ values are -4.998, -4.98, and -4.8. What is the best estimate for $\lim_{x \to 1} g(f(x))$ ?
Answer: -5 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.
The functions $f$ and $g$ are continuous. A table of values near $x=4$ is given. For $x$ values of 3.9, 3.99, and 3.999, the $f(x)$ values are 6.8, 6.98, and 6.998, and the $g(x)$ values are -2.2, -2.02, and -2.002. For $x$ values of 4.001, 4.01, and 4.1, the $f(x)$ values are 7.002, 7.02, and 7.2, and the $g(x)$ values are -1.998, -1.98, and -1.8. What is the best estimate for $\lim_{x \to 4} (f(x) \cdot g(x))$ ?
Answer: -14 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.
A differentiable function $f$ satisfies $f(2) = 8$ . A table of values for the difference quotient $\frac{f(2+h)-f(2)}{h}$ is given. For $h$ values of -0.1, -0.01, and -0.001, the quotient's values are 3.9, 3.99, and 3.999. For $h$ 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 $\lim_{h \to 0} \frac{f(2+h)-f(2)}{h}$ ?
Answer: 4 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 $f(2)$ .
The functions $f$ and $g$ are continuous. A table of values near $x=-3$ is given. For $x$ values of -3.1, -3.01, and -3.001, the $f(x)$ values are 11.7, 11.97, and 11.997, and the $g(x)$ values are 1.9, 1.99, and 1.999. For $x$ values of -2.999, -2.99, and -2.9, the $f(x)$ values are 12.003, 12.03, and 12.3, and the $g(x)$ values are 2.001, 2.01, and 2.1. What is the best estimate for $\lim_{x \to -3} \frac{f(x)}{g(x)}$ ?
Answer: 6 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.
Values of a function $g$ are given for selected values of $x$ near -4. For $x$ values of -4.1, -4.01, and -4.001, the $g(x)$ values are 500, 5000, and 50000. For $x$ values of -3.999, -3.99, and -3.9, the $g(x)$ values are 49999, 4999, and 499. The function is undefined at $x=-4$ . What is the best estimate for $\lim_{x \to -4} g(x)$ ?
Answer: $\infty$ 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 $\infty$ ).
The function $f$ is continuous. A table of values for $f(x)$ near $x=0$ is given. For $x$ values of -0.1, -0.01, and -0.001, the $f(x)$ values are 4.8, 4.98, and 4.998. For $x$ values of 0.001, 0.01, and 0.1, the $f(x)$ values are 5.002, 5.02, and 5.2. What is the best estimate for $\lim_{x \to 0} (e^x \cdot f(x))$ ?
Answer: 5 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.
The function $f$ is continuous. A table of values for $f(x)$ is given: for $x$ values of 10, 100, 1000, and 10000, the corresponding $f(x)$ values are 3.1, 3.01, 3.001, and 3.0001. What is the best estimate for $\lim_{x \to \infty} f(x)$ ?
Answer: 3 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.
A function $f$ is continuous. For $x$ values of 2.9, 2.99, and 2.999, the values for $f(x)$ are -0.1, -0.01, and -0.001. For $x$ values of 3.001, 3.01, and 3.1, the values for $f(x)$ are 0.001, 0.01, and 0.1. We know $f(3)=0$ . What is the best estimate for $\lim_{x \to 3} \frac{\sin(f(x))}{f(x)}$ ?
Answer: 1 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.
Values of a function $f$ are given for selected values of $x$ near 2. The table shows that for $x$ values of 1.9, 1.99, and 1.999, the corresponding $f(x)$ values are 4.71, 4.97, and 4.997. For $x$ values of 2.001, 2.01, and 2.1, the corresponding $f(x)$ values are 5.003, 5.03, and 5.31. The value of $f(2)$ is 7. What is the best estimate for $\lim_{x \to 2} f(x)$ ?
Answer: 5 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.
The functions $f$ and $g$ are continuous. A table of values near $x=4$ is given. For $x$ values of 3.9, 3.99, and 3.999, the $f(x)$ values are 6.8, 6.98, and 6.998, and the $g(x)$ values are -2.2, -2.02, and -2.002. For $x$ values of 4.001, 4.01, and 4.1, the $f(x)$ values are 7.002, 7.02, and 7.2, and the $g(x)$ values are -1.998, -1.98, and -1.8. What is the best estimate for $\lim_{x \to 4} (f(x) + g(x))$ ?
Answer: 5 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.
Values of a function $h$ are given for selected values of $x$ near 3. For $x$ values of 2.9, 2.99, and 2.999, the corresponding $h(x)$ values are -1.5, -1.95, and -1.995. For $x$ values of 3.001, 3.01, and 3.1, the corresponding $h(x)$ values are -3.998, -3.98, and -3.8. The function is undefined at $x=3$ . What is the best estimate for $\lim_{x \to 3^+} h(x)$ ?
Answer: -4 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.
The function $g$ has a removable discontinuity at $x=1$ . For $x$ values of 0.9, 0.99, and 0.999, the $g(x)$ values are -2.19, -2.0199, and -2.001999. For $x$ values of 1.001, 1.01, and 1.1, the $g(x)$ values are -1.997999, -1.9799, and -1.79. The value of $g(1)$ is 5. What is the best estimate for $\lim_{x \to 1} g(x)$ ?
Answer: -2 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.

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