Limits and continuity Knowledge-Based Download PDF

Questions

1. 1.

   What is the value of the limit $\lim_{x \to -\infty} \frac{\sqrt{16x^2 + 7}}{2x - 5}$?

   • A) 2
   • B) -2
   • C) -4
   • D) 4
2. 2.

   What is the value of the one-sided limit $\lim_{x \to -5^+} \frac{|x + 5|}{2x + 10}$?

   • A) $\frac{1}{2}$
   • B) $-\frac{1}{2}$
   • C) 0
   • D) The limit does not exist
3. 3.

   Consider the rational function $f(x) = \frac{x^2 - x - 6}{x^2 - 4}$. Which of the following statements correctly classifies the discontinuities of $f$?

   • A) $f$ has a removable discontinuity at $x = -2$ and an infinite discontinuity at $x = 2$.
   • B) $f$ has an infinite discontinuity at $x = -2$ and a removable discontinuity at $x = 2$.
   • C) $f$ has removable discontinuities at both $x = -2$ and $x = 2$.
   • D) $f$ has jump discontinuities at both $x = -2$ and $x = 2$.
4. 4.

   For what value of the constant $k$ is the function $f(x) = \begin{cases} kx^2 - 3, & x \le 2 \\ 2x + k, & x > 2 \end{cases}$ continuous at $x = 2$?

   • A) $\frac{7}{3}$
   • B) $\frac{1}{3}$
   • C) 7
   • D) $\frac{7}{5}$
5. 5.

   Let $f$ be defined by $f(x) = \begin{cases} \frac{\sqrt{x+4} - 2}{x}, & x \neq 0 \\ k, & x = 0 \end{cases}$. For what value of $k$ is $f$ continuous at $x = 0$?

   • A) $\frac{1}{2}$
   • B) $\frac{1}{4}$
   • C) 2
   • D) 4