BC Calculus
Chapter 2 Multiple Choice Review Questions

  1. \(\displaystyle \displaystyle \lim_{x\to\infty} \frac{5x^3 + 27}{20x^2 + 10x + 9} = \)
    1. \( \infty\)
    2. \(\frac{1}{4} \)
    3. 3
    4. 0
    5. 1
    2. Solution A
  2. \( \displaystyle \lim_{x \to 0} \frac{\tan{x}}{x}= \)
    1. 0
    2. 1
    3. \(\pi\)
    4. \( \infty\)
    5. Nonexistent because left and right-handed limits disagree
    3. Solution B
  3. \( \displaystyle \lim_{x \to 0} \frac{\sin{(2x)}}{x} = \)
    1. 1
    2. 2
    3. \(\frac{1}{2} \)
    4. 0
    5. \(\infty\)
    4. Solution B
  4. \( \displaystyle \lim_{x \to 0} \sin{\frac{1}{x}} = \)
    1. \( \infty \)
    2. 1
    3. nonexistent due to oscillation
    4. -1
    5. none of these
    5. Solution C
  5. \( \displaystyle \lim_{x \to 0} \tan{ \left( \frac{\pi x}{ x} \right) } = \)
    1. \( \frac{1}{\pi}\)
    2. 0
    3. 1
    4. \(\pi\)
    5. \(\infty\)
    6. Solution B
  6. Let \(f(x) = \begin{cases} x^2 - 1, & \text{if } x \neq 1 \\ 4, & \text{if }x = 1 \end{cases}\). Which of the following statements are true?
    1. \( \displaystyle \lim_{x \to 1} f(x)\) exists.
    2. \(f(1)\) exists.
    3. \(f\) is continuous at \(x = 1\).

    1. only I
    2. only II
    3. I and II
    4. none of them
    5. all of them
    7. Solution C
  7. If \(f(x) = \begin{cases} \frac{x^2 - x}{2x}, & \text{if } x \neq 0 \\ k, & \text{if }x = 0 \end{cases}\), then if \(f\) is continuous at \(x = 0\text{, }k = \)
    1. -1
    2. \( -\frac{1}{2}\)
    3. 0
    4. \(\frac{1}{2} \)
    5. 1
    8. Solution B
  8. Supppose\( \displaystyle f(x) = \begin{cases} \frac{3x(x-1)}{x^2 - 3x + 2}, & \text{if } x \neq \{1, 2\} \\ -3, & \text{if } x = 1 \\ 4, & \text{if } x = 2 \end{cases}\). Then \(f(x)\) is continous...
    1. except at \(x = 1\)
    2. except at \(x = 2\)
    3. except at \(x = 1\text{ or }2\)
    4. except at \( x = 0, 1 \text{, or }2\)
    5. at each real number

    9. Solution B

  9. \( \displaystyle \lim_{x \to \infty} \frac {\sin{x}}{x^2 - 3x + 1} = \)
    1. 0
    2. 1
    3. \( \pi \)
    4. \( \infty \)

    Solution A

  10. Given \( \displaystyle f(x) \) is defined on \( [-1, 12] \) and that \( f(-1) = 4,\) and \(f(12) = 10, \) which of the following must be true?
    1. \( \displaystyle \lim_{x \to -1} = 4 \)
    2. \( \displaystyle \lim_{x \to 0} f(x) = f(0) \)
    3. \( \displaystyle \left. \exists m \in [-1, 12] \right| f(m) = 6 \)
    4. The average rate of change of \( f(x) \) on \( [-1, 12] \) is \( \displaystyle \frac {6}{13} \)

    Solution D. Note that the stem of the question never states that \( f \) is continuous.

  11. Which of the following is a statement of the Sandwich Theorem?
    1. If \( p(x) \lt k(x) \lt q(x)\) on x \( \in [a, b],\) \( c \in (a, b), \) and \( \displaystyle \lim_{x \to c} p(x) = \lim_{x \to c} q(x) = L, \) then \( \displaystyle \lim_{x \to c} k(x) = L.\)
    2. If \( p(x) \le k(x) \le q(x)\) on x \( \in [a, b],\) \( c \in [a, b], \) and \( \displaystyle \lim_{x \to c} p(x) = \lim_{x \to c} q(x) = L, \) then \( \displaystyle \lim_{x \to c} k(x) = L.\)
    3. If \( p(x) \le k(x) \le q(x)\) on x \( \in [a, b],\) then \( \displaystyle \left. \exists c \in (a, b) \right| \lim_{x \to c} k(x) = \lim_{x \to c} p(x) \).
    4. None of these.

    Solution B. If you got this wrong, look up and memorize the Sandwich Theorem.

  12. Given \( p(x) \) is any polynomial function, \( \displaystyle \lim_{x \to \infty} \frac {\cos{x}}{p(x)} = \)
    1. 0
    2. 1
    3. \( \pi \)
    4. Does Not Exist

    Solution A