AP Calculus AB Unit 1: Limits and Continuity Practice Exam

Assessment for Unit 1: Limits and Continuity

Estimated Time: 60 minutes

Exam Details

Questions: 47 total (44 multiple choice, 3 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 table below gives values for a continuous function $f$ at selected values of $x$. | $x$ | $1.9$ | $1.99$ | $1.999$ | $2.001$ | $2.01$ | $2.1$ | | :--- | :--- | :--- | :--- | :--- | :--- | :--- | | $f(x)$ | $3.61$ | $3.9601$ | $3.9960$ | $4.0040$ | $4.0401$ | $4.41$ | Based on the data in the table, which of the following best estimates the instantaneous rate of change of $f$ at $x=2$?

(A)2(B)4(C)4.004(D)8

2. The function $H(t)$ represents the heat energy, in joules, contained in a system at time $t$ seconds. Which of the following limits represents the instantaneous rate of change of the heat energy, in joules per second, at time $t=5$?

(A)$\lim_{t \to 5} \frac{H(t) - H(0)}{t - 5}$(B)$\lim_{h \to 0} \frac{H(5+h) - H(5)}{h}$(C)$\frac{H(5) - H(0)}{5 - 0}$(D)$\lim_{h \to 0} \frac{H(5+h) - H(5)}{5}$

3. Let $f$ be a function defined for all real numbers. The average rate of change of $f$ over the interval $[a, a+h]$ is given by the expression $\frac{f(a+h) - f(a)}{h}$. To determine the instantaneous rate of change of $f$ at $x=a$, we evaluate the limit of this expression as $h \to 0$. Which of the following best explains why this limit is necessary?

(A)Because the function $f$ is discontinuous at $x=a$ and the limit bridges the gap.(B)Because the expression for the average rate of change is always equal to 0 when $h$ is small.(C)Because at $h=0$, the change in the independent variable is zero, making the difference quotient undefined (form $\frac{0}{0}$).(D)Because the instantaneous rate of change represents the area under the curve, which requires a limit of a sum.

4. Let $f$ be the function defined by $f(x) = \frac{x^2 - x - 6}{x^2 - 9}$. Which of the following correctly describes the points of discontinuity of the graph of $f$?

(A)There is a removable discontinuity at $x=3$ and an infinite discontinuity at $x=-3$.(B)There is an infinite discontinuity at $x=3$ and a removable discontinuity at $x=-3$.(C)There are infinite discontinuities at both $x=3$ and $x=-3$.(D)There are removable discontinuities at both $x=3$ and $x=-3$.

5. Let $f$ be the piecewise function defined below. $$f(x) = \begin{cases} 2x + 3 & \text{for } x < 1 \\ 5 & \text{for } x = 1 \\ x^2 + 2 & \text{for } x > 1 \end{cases}$$ Which of the following statements correctly classifies the discontinuity of $f$ at $x=1$?

(A)The function $f$ has a removable discontinuity at $x=1$ because $f(1)$ is defined.(B)The function $f$ has a removable discontinuity at $x=1$ because $\lim_{x \to 1} f(x)$ exists.(C)The function $f$ has a jump discontinuity at $x=1$ because $\lim_{x \to 1^-} f(x) \neq \lim_{x \to 1^+} f(x)$.(D)The function $f$ has an infinite discontinuity at $x=1$ because $f(x)$ is undefined at $x=1$.

6. The function $f$ has a discontinuity at $x=c$. If $\lim_{x \to c} f(x)$ exists and is a finite real number, which of the following must be true regarding the type of discontinuity?

(A)The discontinuity is a jump discontinuity.(B)The discontinuity is a removable discontinuity.(C)The discontinuity is an infinite discontinuity.(D)The type of discontinuity cannot be determined without knowing the value of $f(c)$.

7. Let $f$ be the function defined by: $$f(x) = \begin{cases} e^{x-1} & \text{for } x < 1 \\ 2 & \text{for } x = 1 \\ x^2 & \text{for } x > 1 \end{cases}$$ Which of the following statements correctly justifies conclusions about the continuity of $f$ at $x=1$?

(A)$f$ is continuous at $x=1$ because $\lim_{x \to 1} f(x) = 1$ and $f(1)$ is defined.(B)$f$ is not continuous at $x=1$ because $f(1)$ is not defined.(C)$f$ is not continuous at $x=1$ because $\lim_{x \to 1} f(x)$ does not exist.(D)$f$ is not continuous at $x=1$ because $\lim_{x \to 1} f(x) \neq f(1)$.

Refer to the figure below.

8. The graph of a function $f$ is shown in the figure above. Which of the following statements is true regarding the continuity of $f$ at $x=3$?

(A)$f$ is not continuous at $x=3$ because $f(3)$ is undefined.(B)$f$ is not continuous at $x=3$ because $\lim_{x \to 3} f(x)$ does not exist.(C)$f$ is not continuous at $x=3$ because $\lim_{x \to 3} f(x) \neq f(3)$.(D)$f$ is continuous at $x=3$ because the graph is defined on both sides of $x=3$.

9. Let $f$ be the function defined by: $$f(x) = \begin{cases} \frac{\sin(2x)}{x} & \text{for } x < 0 \\ x + k & \text{for } x \ge 0 \end{cases}$$ For what value of $k$ is $f$ continuous at $x=0$?

(A)0(B)1(C)2(D)Undefined

10. Let $f$ be the function defined by $f(x) = \ln(16 - x^2)$. Which of the following is the largest interval on which $f$ is continuous?

(A)$(4, \infty)$(B)$[-4, 4]$(C)$(-4, 4)$(D)$(-\infty, -4) \cup (4, \infty)$

11. Consider the piecewise function $g(x)$ defined below. $$g(x) = \begin{cases} \frac{\sin(3x)}{x} & \text{for } x < 0 \\ k - 2x & \text{for } x \ge 0 \end{cases}$$ For what value of the constant $k$ is the function $g$ continuous on the interval $(-\infty, \infty)$?

(A)0(B)1(C)3(D)There is no such value of $k$.

12. The function $h$ is defined by $h(x) = \frac{x^2}{2\cos(x) - 1}$. On which of the following sets of intervals is $h$ continuous for $x \in [0, \pi]$?

(A)$[0, \pi]$(B)$[0, \frac{\pi}{3}) \cup (\frac{\pi}{3}, \pi]$(C)$[0, \frac{\pi}{6}) \cup (\frac{\pi}{6}, \pi]$(D)$(\frac{\pi}{3}, \pi]$

13. Let $f$ be the function defined by $f(x) = \frac{2x^2 - 5x - 3}{x - 3}$ for all $x \neq 3$. If $f$ is continuous at $x = 3$, what is the value of $f(3)$?

(A)0(B)1(C)7(D)The limit does not exist.

14. Consider the piecewise function $g(x)$ defined below, where $k$ is a constant. $$ g(x) = \begin{cases} kx + 5 & \text{for } x < 2 \\ x^2 - kx + 3 & \text{for } x \ge 2 \end{cases} $$ For what value of $k$ is the function $g$ continuous at $x = 2$?

(A)0.5(B)1(C)2(D)No such value of $k$ exists.

15. The function $h$ is defined by $h(x) = \frac{\sqrt{x+7}-3}{x-2}$ for $x \neq 2$. Which of the following statements is true regarding the graph of $h$ at $x=2$?

(A)The graph has a vertical asymptote at $x=2$.(B)The graph has a jump discontinuity at $x=2$.(C)The discontinuity at $x=2$ can be removed by defining $h(2) = \frac{1}{6}$.(D)The discontinuity at $x=2$ can be removed by defining $h(2) = 0$.

16. The function $f$ is defined by $f(x) = \frac{x(x-1)}{(x-1)(x+3)}$. Which of the following correctly identifies the vertical asymptote of $f$ and provides the correct limit justification?

(A)The line $x = 1$ is a vertical asymptote because $\lim_{x \to 1} f(x) = \infty$.(B)The line $x = -3$ is a vertical asymptote because $\lim_{x \to -3^+} f(x) = -\infty$.(C)The line $x = -3$ is a vertical asymptote because $\lim_{x \to -3^+} f(x) = \infty$.(D)The line $y = 1$ is a vertical asymptote because $\lim_{x \to \infty} f(x) = 1$.

17. The table below gives values of a function $f$ for selected values of $x$ near $5$. | $x$ | $4.9$ | $4.99$ | $4.999$ | $5.001$ | $5.01$ | $5.1$ | | :--- | :--- | :--- | :--- | :--- | :--- | :--- | | $f(x)$ | $100$ | $10,000$ | $1,000,000$ | $-1,000,000$ | $-10,000$ | $-100$ | Based on the data in the table, which of the following conclusions about the graph of $f$ is best supported?

(A)The graph of $f$ has a horizontal asymptote at $y=5$.(B)The graph of $f$ has a vertical asymptote at $x=5$.(C)The graph of $f$ has a removable discontinuity at $x=5$.(D)The limit $\lim_{x \to 5} f(x)$ exists and equals $0$.

18. Let $f$ be the function defined by $f(x) = \frac{6x - 2}{\sqrt{9x^2 + 5}}$. Which of the following describes all horizontal asymptotes to the graph of $f$?

(A)$y = \frac{2}{3}$ only(B)$y = 2$ only(C)$y = 2$ and $y = -2$(D)$y = \frac{2}{3}$ and $y = -\frac{2}{3}$

19. Which of the following is the value of $\lim_{x \to \infty} \frac{10x^3 - 4x + 7}{e^{0.5x} + x^2}$?

(A)0(B)10(C)20(D)$\infty$

20. Let $f$ be the function defined by $f(x) = \frac{kx^n + 3}{4x^3 - 5x}$, where $k$ is a nonzero constant and $n$ is a positive integer. For which values of $k$ and $n$ does the graph of $f$ have a horizontal asymptote at $y = -\frac{1}{2}$?

(A)$n = 1$ and $k = -2$(B)$n = 3$ and $k = -2$(C)$n = 3$ and $k = -\frac{1}{2}$(D)$n = 4$ and $k = -2$

21. Let $f$ be the function defined by $f(x) = \begin{cases} 2x - 4 & \text{for } x < 1 \\ x + 1 & \text{for } x \ge 1 \end{cases}$. We verify that $f(0) = -4$ and $f(2) = 3$. Although $f(0) < 0$ and $f(2) > 0$, there is no value $c$ in the interval $(0, 2)$ such that $f(c) = 0$. Which of the following statements best explains why the Intermediate Value Theorem does not guarantee the existence of such a value $c$?

(A)The function $f$ is not continuous on the interval $[0, 2]$.(B)The interval $(0, 2)$ is an open interval, but the theorem requires a closed interval.(C)The function $f$ is not differentiable at $x=1$.(D)The value 0 is not between $f(0)$ and $f(2)$.

22. Let $h$ be the function defined by $h(x) = e^x + 2x - 4$. The Intermediate Value Theorem guarantees that there is a value $c$ such that $h(c) = 0$ in which of the following intervals?

(A)$(-1, 0)$(B)$(0, 1)$(C)$(1, 2)$(D)$(2, 3)$

23. The table below gives selected values for a function $f$ around $x=2$. | $x$ | $1.9$ | $1.99$ | $1.999$ | $2.001$ | $2.01$ | $2.1$ | | :--- | :--- | :--- | :--- | :--- | :--- | :--- | | $f(x)$ | $3.8$ | $3.98$ | $3.998$ | $4.002$ | $4.02$ | $4.2$ | Based on the values in the table, which of the following best represents $\lim_{x\to 2} f(x)$?

(A)2(B)3.998(C)4(D)The limit does not exist.

Refer to the figure below.

24. The graph of the function $g$ is shown in the figure above. The graph consists of a line segment for $x < 3$ and a curve for $x > 3$. The point $(3, 1)$ is plotted as a solid circle, and there is an open circle at $(3, 5)$. Which of the following statements is true?

(A)$\lim_{x\to 3} g(x) = 1$(B)$\lim_{x\to 3} g(x) = 5$(C)$\lim_{x\to 3} g(x) = 6$(D)$\lim_{x\to 3} g(x)$ does not exist.

25. Which of the following statements best describes the meaning of $\lim_{x\to c} f(x) = L$?

(A)For all $x=c$, the value of $f(x)$ is equal to $L$.(B)The function $f$ is continuous at $x=c$ and $f(c)=L$.(C)The values of $f(x)$ can be made arbitrarily close to $L$ by taking $x$ sufficiently close to $c$, but not equal to $c$.(D)The function $f$ is defined for all real numbers, and the value of $f(x)$ eventually equals $L$.

Refer to the figure below.

26. The graph of the function $f$ is shown in the figure above. The graph contains a removable discontinuity at $x = 2$, where the coordinates of the open circle are $(2, 3)$. The point $(2, 1)$ is solid and included on the graph. Which of the following is the value of $\lim_{x \to 2} f(x)$?

(A)1(B)2(C)3(D)The limit does not exist.

Refer to the figure below.

27. The graph of the function $f$ is shown in the figure above. Which of the following statements is true regarding the limits of $f$ at $x = -1$?

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

Refer to the figure below.

28. The graph of the function $g$, defined on the closed interval $[-2, 6]$, is shown above. The graph consists of line segments and a semi-circle. It has a removable discontinuity at $x=1$ and a jump discontinuity at $x=3$. For which of the following values of $c$ does $\lim_{x \to c} g(x)$ NOT exist?

(A)1 only(B)3 only(C)1 and 3 only(D)1, 3, and 5

29. [Skill: 2.B | Topic: 1.4] The function $f$ is defined for all real numbers except $x = 4$. The table below gives values of $f(x)$ at selected values of $x$. | $x$ | 3.9 | 3.99 | 3.999 | 4.001 | 4.01 | 4.1 | | :--- | :--- | :--- | :--- | :--- | :--- | :--- | | $f(x)$ | 5.800 | 5.980 | 5.998 | 6.002 | 6.020 | 6.200 | Based on the data in the table, which of the following is the best estimate for $\lim_{x \to 4} f(x)$?

(A)4(B)6(C)6.002(D)The limit does not exist.

30. [Skill: 2.B | Topic: 1.4] The function $g$ is defined for all real numbers. The table below gives values of $g(x)$ for values of $x$ near $-2$. | $x$ | $-2.1$ | $-2.01$ | $-2.001$ | $-2$ | $-1.999$ | $-1.99$ | $-1.9$ | | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | | $g(x)$ | 4.9 | 4.99 | 4.999 | 8 | 4.999 | 4.99 | 4.9 | Based on the table, which of the following statements is true regarding $\lim_{x \to -2} g(x)$?

(A)The limit is 5.(B)The limit is 8.(C)The limit is 4.999.(D)The limit does not exist.

31. [Skill: 2.B | Topic: 1.4] The table below shows values of a function $h(x)$ for selected values of $x$ near $0$. | $x$ | $-0.1$ | $-0.01$ | $-0.001$ | $0.001$ | $0.01$ | $0.1$ | | :--- | :--- | :--- | :--- | :--- | :--- | :--- | | $h(x)$ | $-1.5$ | $-1.05$ | $-1.005$ | $3.005$ | $3.05$ | $3.5$ | Which of the following best describes $\lim_{x \to 0} h(x)$?

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

32. Given that $\lim_{x \to 2} f(x) = -4$ and $\lim_{x \to 2} g(x) = 5$, what is the value of $\lim_{x \to 2} \sqrt{g(x) - 2f(x)}$ ?

(A)1(B)$\sqrt{3}$(C)3(D)$\sqrt{13}$

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

(A)-1/9(B)0(C)1/9(D)The limit does not exist.

34. Evaluate $\lim_{x \to 2} \frac{x^2 - 4}{x^2 - 5x + 6}$.

(A)-4(B)0(C)1(D)4

35. Evaluate $\lim_{x \to 9} \frac{3 - \sqrt{x}}{x - 9}$.

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

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

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

37. Evaluate $\lim_{x \to 2} \frac{\sqrt{x+7} - 3}{x - 2}$.

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

38. Evaluate $\lim_{x \to 0} \frac{\frac{1}{x+5} - \frac{1}{5}}{x}$.

(A)-$\frac{1}{25}$(B)0(C)$\frac{1}{25}$(D)Nonexistent

39. Let $f$ be a function such that $2x - 1 \le f(x) \le x^2$ for all $x$ in the interval $(0, 2)$. Which of the following statements is true regarding $\lim_{x \to 1} f(x)$?

(A)The limit is 0.(B)The limit is 1.(C)The limit is 2.(D)The limit does not exist because $f(x)$ is squeezed between two different values.

40. Which of the following gives the value of $\lim_{x \to 0} \left( x^4 \cos\left(\frac{3}{x}\right) \right)$ ?

(A)0(B)1(C)3(D)Nonexistent

41. The function $f$ is continuous for all real numbers, and selected values of $f(x)$ are given in the table below. | $x$ | $1.9$ | $1.99$ | $1.999$ | $2.001$ | $2.01$ | $2.1$ | | :--- | :--- | :--- | :--- | :--- | :--- | :--- | | $f(x)$ | $4.8$ | $4.98$ | $4.998$ | $5.002$ | $5.02$ | $5.2$ | Let $g$ be the function defined by $g(x) = \frac{|x-2|}{x-2}$. Which of the following statements is true regarding $\lim_{x \to 2} (f(x) \cdot g(x))$ ?

(A)The limit is $5$.(B)The limit is $-5$.(C)The limit is $0$.(D)The limit does not exist.

42. The function $h$ is defined by $h(x) = \frac{x^2 - 9}{x - 3}$. Which of the following tables best represents the behavior of the graph of $h$ near $x=3$ ?

(A)x | 2.9 | 2.99 | 3.01 | 3.1 h(x) | 5.9 | 5.99 | 6.01 | 6.1(B)x | 2.9 | 2.99 | 3.01 | 3.1 h(x) | 0.9 | 0.99 | 1.01 | 1.1(C)x | 2.9 | 2.99 | 3.01 | 3.1 h(x) | -6.1 | -6.01 | 6.01 | 6.1(D)x | 2.9 | 2.99 | 3.01 | 3.1 h(x) | Undefined | Undefined | Undefined | Undefined

43. Let $f$ be a function that is continuous on the closed interval $[0, 5]$. The table below gives values of $f$ at selected values of $x$. | $x$ | $0$ | $2$ | $5$ | | :--- | :--- | :--- | :--- | | $f(x)$ | $-3$ | $4$ | $-2$ | Which of the following statements must be true?

(A)The equation $f(x) = 0$ has exactly two solutions in the interval $(0, 5)$.(B)There are at least two values $c$ in the interval $(0, 5)$ such that $f(c) = 3$.(C)The maximum value of $f$ on the interval $[0, 5]$ is $4$.(D)There exists a number $c$ in the interval $(0, 5)$ such that $f'(c) = 0$.

44. Let $f$ be the function defined by $f(x) = \frac{x^2 + 3x - 10}{x^2 - 4}$. Which of the following statements correctly identifies a vertical asymptote of the graph of $f$ and provides a valid justification?

(A)$x = 2$ is a vertical asymptote because $f(2)$ is undefined.(B)$x = -2$ is a vertical asymptote because $\lim_{x \to -2^+} f(x) = \infty$.(C)Both $x = 2$ and $x = -2$ are vertical asymptotes because the denominator of $f$ is zero at these values.(D)Neither $x = 2$ nor $x = -2$ is a vertical asymptote because $\lim_{x \to \infty} f(x) = 1$.

B. Short Answer Questions (3 Questions)

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

Question 45:

Question 46:

Question 47:

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