# BC Calculus Chapter 2 Multiple Choice Review Questions

• Class: 5H: BC Calculus
• Author: Peter Atlas
• Text: Calculus Finney, Demana, Waits, Kennedy

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
3. $$\displaystyle \lim_{x \to 0} \frac{\tan{x}}{x}=$$
1. 0
2. 1
3. $$\pi$$
4. $$\infty$$
5. Nonexistent because left and right-handed limits disagree
4. Solution B
5. $$\displaystyle \lim_{x \to 0} \frac{\sin{(2x)}}{x} =$$
1. 1
2. 2
3. $$\frac{1}{2}$$
4. 0
5. $$\infty$$
6. Solution B
7. $$\displaystyle \lim_{x \to 0} \sin{\frac{1}{x}} =$$
1. $$\infty$$
2. 1
3. nonexistent due to oscillation
4. -1
5. none of these
8. Solution C
9. $$\displaystyle \lim_{x \to 0} \tan{ \left( \frac{\pi x}{ x} \right) } =$$
1. $$\frac{1}{\pi}$$
2. 0
3. 1
4. $$\pi$$
5. $$\infty$$
10. Solution B
11. 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
12. Solution C
13. 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
14. Solution B
15. 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
16. Solution B

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

Solution A

18. 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.

19. 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.

20. 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