Limits and continuity Knowledge-Based Practice

Questions

1. 1.

   What is the value of the one-sided limit $\lim_{x \to 3^-} \frac{x^2 - 2x - 3}{|x - 3|}$?

   • A) 4
   • B) -1
   • C) -4
   • D) The limit does not exist
2. 2.

   What is the value of $\lim_{x \to 0} \frac{1 - \cos(4x)}{x^2}$?

   • A) 4
   • B) 8
   • C) 16
   • D) 2
3. 3.

   Let $f$ be a continuous function on the closed interval $[0, 4]$ with values given by $f(0) = 1$, $f(1) = 5$, $f(2) = 2$, $f(3) = 6$, and $f(4) = 3$. According to the Intermediate Value Theorem, what is the guaranteed minimum number of solutions to the equation $f(x) = 4$ in the open interval $(0, 4)$?

   • A) 4
   • B) 3
   • C) 2
   • D) 1
4. 4.

   Let $f$ be the function defined by $f(x) = \begin{cases} ax^2 + 1, & x \le 2 \\ 2x - a, & x > 2 \end{cases}$. For what value of the constant $a$ is $f$ continuous for all real numbers?

   • A) $a = \frac{1}{3}$
   • B) $a = \frac{1}{2}$
   • C) $a = \frac{5}{3}$
   • D) $a = \frac{3}{5}$
5. 5.

   Let the function $f$ be defined by $f(x) = \begin{cases} \frac{\sqrt{x+7} - 3}{x - 2}, & x \neq 2 \\ c, & x = 2 \end{cases}$. For what value of the constant $c$ is $f$ continuous at $x = 2$?

   • A) $\frac{1}{6}$
   • B) $\frac{1}{3}$
   • C) $6$
   • D) $0$

Answers and step-by-step solutions appear after you finish the test.