AP Calculus BC Unit 1: Limits and Continuity Practice Exam

Assessment for Unit 1: Limits and Continuity

Estimated Time: 60 minutes

Exam Details

Questions: 46 total (44 multiple choice, 2 short answer)
Time: 60 minutes (recommended)
Topics Covered: Unit 1: Limits and Continuity
Format: Multiple Choice, Short Answer questions matching real exam difficulty
Scoring: Instant results with detailed explanations

A. Multiple-Choice Questions (44 Questions)

Select the one best answer for each question.

1. The function $D(t)$ models the distance traveled by an object, in meters, at time $t$ seconds. The table below lists the average rate of change of $D(t)$ over various time intervals $[3, t]$. | Interval $[3, t]$ | $[3, 3.1]$ | $[3, 3.01]$ | $[3, 3.001]$ | $[3, 3.0001]$ | | :--- | :--- | :--- | :--- | :--- | | Average Rate of Change (m/s) | $9.61$ | $9.06$ | $9.006$ | $9.0006$ | Based on the trend in the table, which of the following is the best estimate for the instantaneous rate of change of $D(t)$ at $t=3$, and what does this value represent?

(A)$9$ m/s; it represents the instantaneous velocity of the object at $t=3$.(B)$9$ m/s; it represents the average velocity of the object over the interval $[3, 3.1]$.(C)$9.61$ m/s; it represents the instantaneous velocity of the object at $t=3$.(D)The value is undefined because the interval width approaches zero.

2. Let $P(t)$ represent the population of a bacteria colony $t$ hours after an experiment begins. Which of the following limits represents the instantaneous rate of change of the population at time $t=4$ hours?

(A)$\lim_{h \to 0} \frac{P(4+h) - P(4)}{4}$(B)$\lim_{h \to 0} \frac{P(4) - P(0)}{4}$(C)$\lim_{h \to 0} \frac{P(4+h) - P(4)}{h}$(D)$\lim_{h \to 4} \frac{P(h) - P(4)}{h}$

3. A student is trying to determine the instantaneous rate of change of a function $f(x)$ at $x=c$ by calculating the average rate of change over the interval $[c, c]$. Why is the average rate of change undefined for this specific interval?

(A)Because the function $f(x)$ is not continuous at $x=c$.(B)Because the change in the independent variable is zero, resulting in division by zero.(C)Because the derivative of $f(x)$ at $x=c$ must be zero.(D)Because an interval must contain at least one integer value.

4. Consider the function $g(x) = \frac{x^2 + 2x - 8}{x - 2}$. Which of the following statements is true regarding the continuity of $g$ at $x=2$?

(A)The function $g$ is continuous at $x=2$.(B)The function $g$ has a jump discontinuity at $x=2$.(C)The function $g$ has a removable discontinuity at $x=2$.(D)The function $g$ has a discontinuity due to a vertical asymptote at $x=2$.

5. The function $h$ is defined by $h(x) = \frac{x-1}{x^2 - 1}$. Which of the following describes the behavior of $h$ at $x = -1$?

(A)The function has a removable discontinuity because $\lim_{x \to -1} h(x)$ exists.(B)The function has a removable discontinuity because $h(-1)$ is undefined.(C)The function has a jump discontinuity because the left and right limits at $x=-1$ are unequal finite numbers.(D)The function has a discontinuity due to a vertical asymptote because $\lim_{x \to -1} h(x)$ is infinite.

6. The function $f$ is given by the graph below. The graph consists of a line segment and a distinct point. Which of the following statements is true regarding the continuity of $f$ at $x=2$?

(A)The function $f$ is continuous at $x=2$ because $f(2)$ is defined.(B)The function $f$ is not continuous at $x=2$ because $\lim_{x \to 2} f(x)$ does not exist.(C)The function $f$ is not continuous at $x=2$ because $f(2)$ is not defined.(D)The function $f$ is not continuous at $x=2$ because $\lim_{x \to 2} f(x) \neq f(2)$.

7. Let $f$ be the function defined by $f(x) = \frac{x^2 + 2x - 8}{x - 2}$. Which of the following statements correctly justifies why $f$ is not continuous at $x=2$?

(A)$f(2)$ is not defined.(B)$\lim_{x \to 2} f(x)$ does not exist.(C)$\lim_{x \to 2^-} f(x) \neq \lim_{x \to 2^+} f(x)$.(D)$\lim_{x \to 2} f(x) \neq f(2)$, even though $f(2)$ is defined.

8. Let $g$ be the piecewise function defined below. $$ g(x) = \begin{cases} \frac{x^2 - 9}{x - 3} & \text{for } x < 3 \\ kx - 5 & \text{for } x \ge 3 \end{cases} $$ For what value of the constant $k$ is the function $g$ continuous at $x=3$?

(A)2(B)3(C)11/3(D)6

9. Let $f$ be the function defined by $f(x) = \frac{x-3}{x^2 - 2x - 3}$. Which of the following describes the set of all values of $x$ for which $f$ is continuous?

(A)All real numbers except $x = 3$(B)All real numbers except $x = -1$(C)All real numbers except $x = 3$ and $x = -1$(D)All real numbers

10. Which of the following functions is continuous on the interval $(0, 4)$?

(A)$f(x) = \tan\left(\frac{\pi x}{2}\right)$(B)$g(x) = \frac{1}{x^2 - 3x + 2}$(C)$h(x) = \ln(x - 2)$(D)$k(x) = \begin{cases} x^2 & \text{for } 0 < x < 2 \\ 2x & \text{for } 2 \le x < 4 \end{cases}$

11. Consider the function $g$ defined by $g(x) = \begin{cases} ax + 2 & \text{for } x < 3 \\ x^2 - a & \text{for } x \ge 3 \end{cases}$. For what value of the constant $a$ is the function $g$ continuous on the interval $(-\infty, \infty)$?

(A)$\frac{7}{4}$(B)$\frac{7}{2}$(C)$7$(D)There is no such value of $a$.

12. The function $f$ is defined by $f(x) = \frac{x^2 + 3x - 10}{x - 2}$ for all $x \neq 2$. If $g$ is a continuous function such that $g(x) = f(x)$ for all $x \neq 2$, what is the value of $g(2)$ ?

(A)0(B)2(C)5(D)7

13. Let $f$ be the function defined below, where $k$ is a constant. $$f(x) = \begin{cases} e^{2x} & x < 0 \\ k \cos(x) + 3 & x \geq 0 \end{cases}$$ For what value of $k$ is $f$ continuous at $x=0$ ?

(A)-2(B)-1(C)1(D)2

14. Consider the function $h(x) = \frac{x^2 - 9}{x^2 + x - 6}$. Which of the following statements correctly identifies the type of discontinuity at $x = -3$ and provides the value necessary to remove it, if possible?

(A)The discontinuity at $x = -3$ is non-removable because the limit does not exist.(B)The discontinuity at $x = -3$ is removable. To make $h$ continuous at $x = -3$, define $h(-3) = 0$.(C)The discontinuity at $x = -3$ is removable. To make $h$ continuous at $x = -3$, define $h(-3) = \frac{6}{5}$.(D)The discontinuity at $x = -3$ is removable. To make $h$ continuous at $x = -3$, define $h(-3) = \frac{3}{2}$.

15. Let $f$ be the function defined by $f(x) = \frac{x^2+3x}{x^2-9}$. Which of the following statements correctly identifies the vertical asymptote(s) of the graph of $f$ and provides a valid justification using limits?

(A)$x = -3$ is a vertical asymptote because $\lim_{x \to -3} f(x) = \infty$.(B)$x = 3$ is a vertical asymptote because $\lim_{x \to 3^+} f(x) = \infty$.(C)$x = 3$ is a vertical asymptote because $\lim_{x \to 3^+} f(x) = 1$.(D)$x = -3$ and $x = 3$ are both vertical asymptotes because the denominator is zero at these values.

16. Consider the function $g(x) = \frac{2-x}{(x-1)^2}$. Which of the following describes the behavior of the graph of $g$ as $x$ approaches 1?

(A)$\lim_{x \to 1} g(x) = \infty$, indicating a vertical asymptote at $x=1$.(B)$\lim_{x \to 1} g(x) = -\infty$, indicating a vertical asymptote at $x=1$.(C)$\lim_{x \to 1^-} g(x) = -\infty$ and $\lim_{x \to 1^+} g(x) = \infty$, indicating a vertical asymptote at $x=1$.(D)The limit exists and is finite, indicating a removable discontinuity at $x=1$.

17. The function $h(x) = \sec(x)$ has a vertical asymptote at $x = \frac{\pi}{2}$. Which of the following one-sided limit statements correctly describes the behavior of $h(x)$ at this asymptote?

(A)$\lim_{x \to \frac{\pi}{2}^-} h(x) = -\infty$(B)$\lim_{x \to \frac{\pi}{2}^-} h(x) = \infty$(C)$\lim_{x \to \frac{\pi}{2}^+} h(x) = \infty$(D)$\lim_{x \to \frac{\pi}{2}^+} h(x) = 0$

18. Let $f$ be the function defined by $f(x) = \frac{5x - 2}{\sqrt{9x^2 + 4}}$. Which of the following describes all horizontal asymptotes to the graph of $f$?

(A)y = 5/3 only(B)y = 5/9 only(C)y = 5/3 and y = -5/3(D)y = 5/9 and y = -5/9

19. Which of the following is the value of $\lim_{x \to \infty} \frac{4e^x + x^{10}}{2e^x - 5x^{10}}$ ?

(A)-1/5(B)0(C)2(D)The limit does not exist.

20. The function $g$ is continuous for all real numbers. The table below gives values of $g(x)$ for selected values of $x$. | $x$ | -10,000 | -1,000 | -100 | ... | 100 | 1,000 | 10,000 | | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | | $g(x)$ | -2.999 | -2.99 | -2.5 | ... | 4.01 | 4.001 | 4.0001 | Based on the data in the table, which of the following statements is most likely true about the graph of $g$?

(A)The graph of g has a horizontal asymptote at y = 4 only.(B)The graph of g has a horizontal asymptote at y = -3 only.(C)The graph of g has horizontal asymptotes at y = -3 and y = 4.(D)The graph of g has no horizontal asymptotes because g(x) is continuous.

21. [Skill: 1.E | Topic: 1.16] The function $g$ is continuous on the closed interval $[0, 9]$. Selected values of $g(x)$ are shown in the table below. | $x$ | 0 | 3 | 6 | 9 | | :--- | :--- | :--- | :--- | :--- | | $g(x)$ | -5 | 2 | -1 | 4 | Based on the values in the table, which of the following statements must be true?

(A)The equation $g(x) = 0$ has no solutions in the interval $[0, 9]$.(B)The equation $g(x) = 0$ has exactly three solutions in the interval $[0, 9]$.(C)The equation $g(x) = 0$ has at least three solutions in the interval $[0, 9]$.(D)The equation $g(x) = 3$ has at least two solutions in the interval $[0, 9]$.

22. [Skill: 2.D | Topic: 1.16] Functions $f$ and $g$ are continuous on the closed interval $[1, 5]$. Selected values for $f(x)$ and $g(x)$ are given in the table below. | $x$ | 1 | 5 | | :--- | :--- | :--- | | $f(x)$ | 6 | 2 | | $g(x)$ | 2 | 5 | Which of the following statements best explains why there must be a value $c$ in the interval $(1, 5)$ such that $f(c) = g(c)$?

(A)Since $f(c) > g(c)$ for all $c$, the graphs never intersect.(B)Since $f(1) > g(1)$ and $f(5) < g(5)$, the Intermediate Value Theorem applied to $h(x) = f(x) - g(x)$ guarantees $h(c) = 0$.(C)Since $f$ is decreasing and $g$ is increasing, they must intersect at exactly one point.(D)The Intermediate Value Theorem guarantees that $f(c) = 4$ and $g(c) = 4$, so $f(c) = g(c)$.

23. [Skill: 3.D | Topic: 1.16] Let $f$ be the function defined by $f(x) = \frac{2x}{x-2}$. Consider the closed interval $[0, 4]$. We observe that $f(0) = 0$ and $f(4) = 4$. However, there is no value $c$ in the interval $(0, 4)$ such that $f(c) = 2$. Why does this not contradict the Intermediate Value Theorem?

(A)The value 2 is not between $f(0)$ and $f(4)$.(B)The function $f$ is not defined for all real numbers.(C)The function $f$ is not continuous on the closed interval $[0, 4]$.(D)The Intermediate Value Theorem only applies to polynomial functions.

24. The table below gives selected values for a continuous function $f$. | $x$ | $1.9$ | $1.99$ | $1.999$ | $2.001$ | $2.01$ | $2.1$ | | :--- | :--- | :--- | :--- | :--- | :--- | :--- | | $f(x)$ | $4.85$ | $4.98$ | $4.999$ | $5.001$ | $5.02$ | $5.15$ | Based on the data in the table, which of the following is the best estimate for $\lim_{x \to 2} f(x)$?

(A)2(B)5(C)4.999(D)The limit does not exist.

25. The graph of the function $f$ is shown in the figure above. The graph consists of a line segment for $x < 3$ and a distinct line segment for $x > 3$. The point $(3, 2)$ is indicated by a solid circle, while the point $(3, 4)$ is indicated by an open circle where the two line segments appear to meet. Which of the following statements is true regarding $\lim_{x \to 3} f(x)$?

(A)$\lim_{x \to 3} f(x) = 2$(B)$\lim_{x \to 3} f(x) = 4$(C)$\lim_{x \to 3} f(x)$ does not exist because $f(3) \neq 4$.(D)$\lim_{x \to 3} f(x)$ does not exist because the graph has a hole at $x=3$.

26. Consider a function $g$. Which of the following expressions represents the statement: "The values of $g(x)$ can be made arbitrarily close to $K$ by taking values of $x$ sufficiently close to $c$ (but not equal to $c$)"?

(A)$g(c) = K$(B)$g(K) = c$(C)$\lim_{x \to K} g(x) = c$(D)$\lim_{x \to c} g(x) = K$

27. [Skill: 2.B | Topic: 1.3] The graph of the function $f$ is shown in the xy-plane. The graph consists of a line segment for $x < 2$ leading to the open point $(2, 3)$, a closed point at $(2, 5)$, and a curve for $x > 2$ starting at the open point $(2, 1)$. Which of the following statements about $f$ is true?

(A)$\lim_{x \to 2} $f(x) = 5(B)$\lim_{x \to 2} $f(x) = 3(C)$\lim_{x \to 2^-} $f(x) = 3(D)$\lim_{x \to 2^+} $f(x) = 5

28. [Skill: 2.B | Topic: 1.3] The graph of the function $g$ is shown. As $x$ approaches 0 from the right, the function $g(x) = \sin(\frac{1}{x})$ oscillates between $-1$ and $1$ with increasing frequency. Based on the graph and the behavior described, what is the estimate of $\lim_{x \to 0^+} g(x)$?

(A)0(B)1(C)The limit does not exist because the function is undefined at x=0.(D)The limit does not exist because the function does not approach a single value.

Refer to the figure below.

29. [Skill: 2.B | Topic: 1.3] The graph of the function $h$ is shown in the figure. Which of the following is the correct value for $\lim_{x \to 3} h(x)$?

(A)2(B)4(C)$\infty$(D)Does Not Exist

30. The function $f$ is defined for all real numbers except $x=2$. The table below gives values of $f(x)$ for selected values of $x$ near 2. | $x$ | 1.9 | 1.99 | 1.999 | 2.001 | 2.01 | 2.1 | | :--- | :--- | :--- | :--- | :--- | :--- | :--- | | $f(x)$ | 5.15 | 5.02 | 5.003 | 4.997 | 4.98 | 4.85 | Based on the data in the table, which of the following is the best estimate for $\lim_{x \to 2} f(x)$?

(A)0(B)2(C)5(D)The limit does not exist.

31. The domain of the function $g$ is all real numbers. The table below lists values of $g(x)$ for several values of $x$ approaching -4. | $x$ | -4.1 | -4.01 | -4.001 | -4 | -3.999 | -3.99 | -3.9 | | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | | $g(x)$ | 2.9 | 2.99 | 2.999 | 8 | -5.001 | -5.01 | -5.1 | Which of the following statements is best supported by the data in the table?

(A)$\lim_{x \to -4} g(x) = 8$(B)$\lim_{x \to -4} g(x) = 3$(C)$\lim_{x \to -4} g(x) = -5$(D)$\lim_{x \to -4} g(x)$ does not exist.

32. Let $h$ be a function defined for all $x \neq 0$. The table below shows values of $h(x)$ for selected values of $x$ near 0. | $x$ | -0.1 | -0.01 | -0.001 | 0.001 | 0.01 | 0.1 | | :--- | :--- | :--- | :--- | :--- | :--- | :--- | | $h(x)$ | 2.9552 | 2.9995 | 2.9999 | 2.9999 | 2.9995 | 2.9552 | Based on the table, which of the following is the best estimate for $\lim_{x \to 0} h(x)$?

(A)0(B)1(C)3(D)The limit does not exist.

33. Evaluate $\lim_{x \to 4} \frac{\sqrt{x+5}-3}{x-4}$.

(A)0(B)$\frac{1}{6}$(C)1(D)Nonexistent

34. Evaluate $\lim_{h \to 0} \frac{\frac{1}{3+h} - \frac{1}{3}}{h}$.

(A)-$\frac{1}{9}$(B)0(C)$\frac{1}{9}$(D)Nonexistent

35. Evaluate $\lim_{x \to -2} \frac{x^2 + 6x + 8}{x^2 - 4}$.

(A)-$\frac{1}{2}$(B)0(C)$\frac{1}{2}$(D)1

36. Evaluate $\lim_{x \to 2} \frac{x^2 - 4}{x^2 + x - 6}$.

(A)0(B)$\frac{4}{5}$(C)1(D)Nonexistent

37. Which of the following is the value of $\lim_{x \to 5} \frac{\sqrt{x+4} - 3}{x - 5}$?

(A)0(B)$\frac{1}{6}$(C)1(D)6

38. Evaluate $\lim_{x \to 0} \frac{\frac{1}{x+3} - \frac{1}{3}}{x}$.

(A)-$\frac{1}{9}$(B)0(C)$\frac{1}{9}$(D)Nonexistent

39. Evaluate $\lim_{x \to 4} \frac{\sqrt{2x+1} - 3}{x - 4}$.

(A)0(B)$\frac{1}{6}$(C)$\frac{1}{3}$(D)The limit does not exist.

40. Let $g$ be a function defined for all real numbers $x \neq 0$ such that $-x^2 \leq g(x) \leq x^2$. Which of the following statements must be true regarding $\lim_{x \to 0} g(x)$?

(A)The limit is 0 because of the Squeeze Theorem.(B)The limit is 1 because $\frac{g(x)}{x^2}$ approaches 1.(C)The limit does not exist because the function oscillates near $x=0$.(D)The limit cannot be determined without an explicit formula for $g(x)$.

41. Let $f$, $g$, and $h$ be functions defined for all $x \neq 2$ such that $g(x) \leq f(x) \leq h(x)$. If $g(x) = -x^2 + 4x - 1$ and $h(x) = e^{x-2} + 2$, what is $\lim_{x \to 2} f(x)$?

(A)1(B)3(C)4(D)The limit cannot be determined from the given information.

42. The function $f$ is defined by $f(x) = x^4 \sin\left(\frac{1}{x^2}\right)$ for all $x \neq 0$. Which of the following inequalities provides sufficient reasoning to determine $\lim_{x \to 0} f(x)$ using the Squeeze Theorem?

(A)$-1 \leq f(x) \leq 1$(B)$-x^4 \leq f(x) \leq x^4$(C)$1 - x^4 \leq f(x) \leq 1 + x^4$(D)$0 \leq f(x) \leq x^4$

43. The table below lists selected values of a function $f$ near $x = 3$. | $x$ | $2.9$ | $2.99$ | $2.999$ | $3.001$ | $3.01$ | $3.1$ | | :--- | :--- | :--- | :--- | :--- | :--- | :--- | | $f(x)$ | $4.8$ | $4.98$ | $4.998$ | $5.002$ | $5.02$ | $5.2$ | Based on the data in the table, which of the following statements is best supported?

(A)$\lim_{x \to 3} f(x) = 5$(B)$\lim_{x \to 3} f(x) = 3$(C)$\lim_{x \to 5} f(x) = 3$(D)The limit $\lim_{x \to 3} f(x)$ does not exist because $f(3)$ is not given in the table.

Refer to the figure below.

44. The graph of the function $g$ is shown in the figure above. Which of the following equations correctly describes the limit behavior of $g$ at $x = -2$?

(A)$\lim_{x \to -2} g(x) = 1$(B)$\lim_{x \to -2} g(x) = 3$(C)$\lim_{x \to -2} g(x) = -2$(D)$\lim_{x \to -2} g(x)$ does not exist.

B. Short Answer Questions (2 Questions)

Answer all parts of each question. Answers must be in essay form. Outlines or lists alone are not acceptable.

Question 45:

Question 46:

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