Connecting Multiple Representations of Limits - AP Calculus AB

A table of values for a function $f(x)$ shows that for $x$ values of $-3.1, -3.01, -3.001$ , the corresponding $f(x)$ values are $5.2, 5.02, 5.002$ . Which of the following limit statements is best supported by this numerical evidence?
Answer: $\lim_{x \to -3^-} f(x) = 5$ The given $x$ values ( $-3.1, -3.01, -3.001$ ) are all less than -3 and are approaching -3. This constitutes an approach from the left. The corresponding $f(x)$ values are approaching 5. Therefore, the data suggests that the left-sided limit is 5, which is written as ${\lim_{x \to -3^-} f(x) = 5}$ . A is incorrect as the data is for a left-sided approach. C incorrectly swaps the roles of $x$ and $f(x)$ . D is a two-sided limit, and we only have evidence for the left side.
The statement ${\lim_{x \to c} f(x)}$ does not exist. A table of values shows that as $x$ approaches $c$ from the left, $f(x)$ approaches 5. As $x$ approaches $c$ from the right, $f(x)$ approaches -5. This describes what feature on the graph of $f$ at $x=c$ ?
Answer: A jump discontinuity The numerical data indicates that the left-sided limit ( ${\lim_{x \to c^-} f(x) = 5}$ ) and the right-sided limit ( ${\lim_{x \to c^+} f(x) = -5}$ ) both exist but are not equal. This is the definition of a jump discontinuity. The graph 'jumps' from one finite value to another at $x=c$ . A is incorrect because a removable discontinuity has a two-sided limit that exists. B is incorrect because a vertical asymptote involves an infinite limit. C is incorrect because a corner or cusp is a point where the function is continuous but not differentiable; the limit would exist.
The mathematical statement ${\lim_{x \to \infty} f(x) = -3}$ implies which of the following about the graph of $y=f(x)$ ?
Answer: The graph has a horizontal asymptote at $y = -3$ . A limit at infinity describes the end behavior of a function. If the limit of $f(x)$ as $x$ approaches infinity is a finite number $L$ , this means the graph of the function gets arbitrarily close to the horizontal line $y=L$ . Therefore, the graph has a horizontal asymptote at $y = -3$ . A describes the behavior ${\lim_{x \to -3} f(x) = \pm\infty}$ . B describes a removable discontinuity, which is related to a limit at a finite point. D describes the y-intercept, which is $f(0)$ , not a limit at infinity.
Suppose ${\lim_{x \to 0^+} f(x) = 1}$ , ${\lim_{x \to 0^-} f(x) = -1}$ , and $f(0) = 1$ . Which verbal statement accurately describes the function's behavior at $x=0$ ?
Answer: The function has a jump discontinuity at $x=0$ and is continuous from the right at $x=0$ . The left-sided limit (-1) and the right-sided limit (1) are not equal, so the function has a jump discontinuity at $x=0$ . The definition of continuity from the right at a point $c$ is ${\lim_{x \to c^+} f(x) = f(c)}$ . In this case, ${\lim_{x \to 0^+} f(x) = 1}$ and $f(0) = 1$ , so the function is continuous from the right. The function is not continuous from the left because ${\lim_{x \to 0^-} f(x) = -1 \neq f(0)}$ . Thus, B is the most complete and accurate description. A is false due to the jump. C is false because the two-sided limit doesn't exist. D is false because it is not continuous from the left.
The graph of the function $h$ has a jump discontinuity at $x=-2$ . It is observed that as $x$ gets closer to -2 from the left, the $y$ -values get closer to 3. As $x$ gets closer to -2 from the right, the $y$ -values get closer to -1. Which pair of mathematical statements represents this information?
Answer: $\lim_{x \to -2^-} h(x) = 3$ and $\lim_{x \to -2^+} h(x) = -1$ The description 'as $x$ gets closer to -2 from the left, the $y$ -values get closer to 3' corresponds to the left-sided limit ${\lim_{x \to -2^-} h(x) = 3}$ . The description 'as $x$ gets closer to -2 from the right, the $y$ -values get closer to -1' corresponds to the right-sided limit ${\lim_{x \to -2^+} h(x) = -1}$ . Choice C correctly pairs these statements. A incorrectly swaps the inputs and outputs. B is logically impossible, as a limit cannot equal two different values. D makes a conclusion about the function value $h(-2)$ , which is not given in the description of the limits.
The values of a function $f(x)$ are tabulated for values of $x$ near 1. For $x = 0.9, 0.99, 0.999$ , the corresponding $f(x)$ values are $4.81, 4.9801, 4.998001$ . For $x = 1.1, 1.01, 1.001$ , the corresponding $f(x)$ values are $5.21, 5.0201, 5.002001$ . Which limit statement do these numerical values suggest?
Answer: $\lim_{x \to 1} f(x) = 5$ The table of values shows that as $x$ approaches 1 from the left (values like 0.9, 0.99, 0.999), the values of $f(x)$ get closer to 5. As $x$ approaches 1 from the right (values like 1.1, 1.01, 1.001), the values of $f(x)$ also get closer to 5. Since the function approaches the same value from both sides, the limit is 5. B is incorrect because the values are approaching a finite number. C incorrectly reverses the roles of the input and the limit value. D is incorrect because the existence of a limit at a point does not depend on the function's value at that point.
Which description of a function's values provides numerical evidence that ${\lim_{x \to 0} g(x)}$ might not exist due to oscillation?
Answer: As $x$ approaches 0, the values of $g(x)$ alternate between 1 and -1 but do not settle on a single value. A limit fails to exist due to oscillation when the function's values do not approach a single number but instead fluctuate between two or more values as $x$ gets closer to the limit point. The classic example is $g(x) = \sin(1/x)$ near $x=0$ . Choice A describes this behavior. B describes a jump discontinuity, a different reason for a limit not to exist. C describes an infinite limit, another reason for a limit not to exist, but it's not oscillation. D describes a function that is continuous at $x=0$ , so the limit exists.
The statement ${\lim_{x \to 2} f(x) = \infty}$ for a rational function $f(x) = \frac{p(x)}{q(x)}$ most likely implies which of the following analytical conditions?
Answer: $p(2) \neq 0$ and $q(2) = 0$ An infinite limit for a rational function typically occurs at a vertical asymptote. A vertical asymptote at $x=c$ happens when the denominator is zero and the numerator is non-zero at $x=c$ . Therefore, for the limit to be infinite at $x=2$ , we would expect $q(2)=0$ and $p(2) \neq 0$ . A would lead to an indeterminate form $0/0$ , which usually corresponds to a removable discontinuity (a hole). C would result in a limit of 0. D would result in a finite, non-zero limit, and the function would be continuous at $x=2$ .
Which of the following is a verbal description of the mathematical statement ${\lim_{x \to 4} f(x) = 2}$ ?
Answer: As $x$ approaches 4, the values of $f(x)$ can be made arbitrarily close to 2. The statement ${\lim_{x \to 4} f(x) = 2}$ describes the behavior of the function $f(x)$ as $x$ gets close to 4. It means the function's values approach 2. The value of $f(4)$ itself is not relevant to the limit. Choice B correctly states this concept. A is incorrect because the limit of a function as $x$ approaches a point does not depend on the value of the function at that point. C incorrectly swaps the roles of $x$ and $f(x)$ . D is too strong; the values only need to approach 2, not be exactly 2.
Suppose that for a function $f$ , it is known that 'the limit of $f(x)$ as $x$ approaches 2 from the left is 5' and 'the limit of $f(x)$ as $x$ approaches 2 from the right is 5'. Which of the following statements must be true?
Answer: $\lim_{x \to 2} f(x) = 5$ For a two-sided limit to exist, the left-sided limit and the right-sided limit must exist and be equal. Since both the left-sided and right-sided limits as $x$ approaches 2 are equal to 5, the two-sided limit ${\lim_{x \to 2} f(x)}$ must exist and be equal to 5. A is not necessarily true; the value of the function at the point is independent of the limit. B is not necessarily true; for continuity, we would also need to know that $f(2) = 5$ . D is incorrect; asymptotes involve infinite limits, not finite ones.
The concept of a limit is that for ${\lim_{x \to c} f(x) = L}$ , we can make $f(x)$ as close as we want to $L$ by making $x$ sufficiently close to $c$ . Which statement best translates this idea into the context of a graph?
Answer: For any horizontal strip around $y=L$ , there is a vertical strip around $x=c$ such that the graph within the vertical strip stays within the horizontal strip. This question describes the epsilon-delta definition of a limit in graphical terms. 'Making $f(x)$ as close as we want to $L$ ' corresponds to defining a small horizontal strip (an epsilon-neighborhood) around the line $y=L$ . 'By making $x$ sufficiently close to $c$ ' corresponds to finding a vertical strip (a delta-neighborhood) around the line $x=c$ such that the portion of the graph inside this vertical strip (excluding possibly at $x=c$ ) is contained entirely within the horizontal strip. Choice B accurately describes this relationship. A is incorrect; the function value at $c$ is irrelevant. C confuses the limit value $L$ with the slope. D incorrectly swaps the roles of the horizontal and vertical strips.
Consider a function $f(x)$ defined piecewise as $f(x) = x^2$ for $x < 2$ and $f(x) = 3x - 2$ for $x > 2$ . Which pair of statements correctly describes the one-sided limits at $x=2$ ?
Answer: $\lim_{x \to 2^-} f(x) = 4$ and $\lim_{x \to 2^+} f(x) = 4$ To find the left-sided limit ( $x \to 2^-$ ), we use the piece of the function defined for $x < 2$ , which is $f(x) = x^2$ . So, ${\lim_{x \to 2^-} x^2 = 2^2 = 4}$ . To find the right-sided limit ( $x \to 2^+$ ), we use the piece of the function defined for $x > 2$ , which is $f(x) = 3x - 2$ . So, ${\lim_{x \to 2^+} (3x-2) = 3(2) - 2 = 4}$ . Both one-sided limits are equal to 4. The other choices contain incorrect calculations for one or both of the one-sided limits.
Which of the following verbal descriptions corresponds to the mathematical statement ${\lim_{x \to -2^-} f(x) = -\infty}$ ?
Answer: As $x$ approaches -2 from the left, the values of $f(x)$ decrease without bound. The notation $x \to -2^-$ means 'as x approaches -2 from the left' (from values less than -2). The notation $f(x) \to -\infty$ means 'the values of f(x) decrease without bound.' Choice A correctly combines these two interpretations. B describes the right-sided limit, ${\lim_{x \to -2^+} f(x) = -\infty}$ . C and D both describe end behavior (limits at infinity), not the behavior near a finite point.
Which of the following limit notations correctly represents the statement 'The end behavior of the function $g(x)$ is that its values approach 4 as $x$ becomes large in the negative direction'?
Answer: $\lim_{x \to -\infty} g(x) = 4$ The phrase 'as $x$ becomes large in the negative direction' means $x$ is approaching negative infinity ( $x \to -\infty$ ). The phrase 'its values approach 4' means the function's output approaches 4. Therefore, the correct notation is ${\lim_{x \to -\infty} g(x) = 4}$ . A and C describe behavior near the finite value $x=4$ . D describes behavior as $x$ approaches positive infinity and incorrectly states the limit value is -4.
The limit statement ${\lim_{x \to -1} f(x)}$ exists. Which of the following verbal statements about the graph of $f$ cannot be true?
Answer: The graph of $f$ has a jump discontinuity at $x=-1$ . If the limit ${\lim_{x \to -1} f(x)}$ exists, it means that ${\lim_{x \to -1^-} f(x) = \lim_{x \to -1^+} f(x)}$ . A jump discontinuity is defined by the left- and right-sided limits existing but being unequal. Therefore, if the two-sided limit exists, the graph cannot have a jump discontinuity. A, B, and D are all situations where the two-sided limit exists. A hole (removable discontinuity) has a limit that exists but is not equal to the function value. A continuous point has a limit that exists and equals the function value. A sharp corner has a limit that exists and equals the function value (it's continuous but not differentiable).
The graph of a function $y=f(x)$ has a vertical asymptote at $x=5$ . As $x$ approaches 5 from the right, the graph of $f(x)$ increases without bound. Which statement represents this behavior?
Answer: $\lim_{x \to 5^+} f(x) = \infty$ The description 'as $x$ approaches 5 from the right' corresponds to a right-sided limit, denoted by $x \to 5^+$ . The description 'the graph of $f(x)$ increases without bound' means the function's values approach positive infinity. Therefore, the correct notation is ${\lim_{x \to 5^+} f(x) = \infty}$ . A represents the behavior as $x$ approaches 5 from the left. C describes end behavior, which corresponds to a horizontal asymptote, not a vertical one. D represents a two-sided limit, but the description only provides information about the right-sided behavior.
If ${\lim_{x \to 1} f(x) = 4}$ and $f(1)=6$ , which verbal statement best describes the function $f$ at $x=1$ ?
Answer: The function has a removable discontinuity at $x=1$ . A removable discontinuity occurs at $x=c$ when ${\lim_{x \to c} f(x)}$ exists but is not equal to $f(c)$ . Here, the limit as $x$ approaches 1 exists and is 4, but the function value $f(1)$ is 6. This perfectly matches the definition of a removable discontinuity, which would appear as a 'hole' in the graph at $(1, 4)$ , with a point at $(1, 6)$ . A is incorrect because for continuity, the limit must equal the function value. B is incorrect because a jump discontinuity requires the left and right-sided limits to be different. D is incorrect because an infinite discontinuity requires the limit to be infinite.
The statement 'The values of a function $g(x)$ get closer and closer to 7 as $x$ gets arbitrarily close to -1 from either side' is represented by which mathematical notation?
Answer: $\lim_{x \to -1} g(x) = 7$ The verbal description indicates that as the input $x$ approaches -1, the output $g(x)$ approaches 7. This is the definition of a two-sided limit. The notation ${\lim_{x \to -1} g(x) = 7}$ correctly represents this statement. A represents the function value at a point, not the limit. B reverses the roles of the input and the limit value. D describes a situation where the right-sided limit is 7 but the left-sided limit is not, so the two-sided limit would not exist.
Suppose for a function $g$ , we know ${\lim_{x \to 1^+} g(x) = -\infty}$ and ${\lim_{x \to 1^-} g(x) = \infty}$ . Which statement best provides a graphical interpretation of this behavior?
Answer: The graph of $g$ has a vertical asymptote at $x=1$ . When the one-sided limit of a function as $x$ approaches a finite number $c$ is either positive or negative infinity, it signifies the presence of a vertical asymptote at $x=c$ . In this case, as $x$ approaches 1, the function's values are unbounded, indicating a vertical asymptote at $x=1$ . The fact that the limits are different from each side describes the behavior on either side of the asymptote. A describes end behavior, not behavior at a finite point. B describes a discontinuity where both one-sided limits are finite but different. D is incorrect because the function is not continuous where it has an infinite limit.

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