1. Let $\displaystyle f(x) = \begin{cases} x^2-8\,x+18 & \mbox{ if } x \le 4 ; \\ 3\,x-8 & \mbox{ if } x > 4 . \end{cases} $

    Find $\displaystyle \lim_{x\rightarrow 4}{f\left(x\right)}$.

    2 4 Does not exist 0 -2

  2. Find $\displaystyle \lim_{x\rightarrow 0}{{{\left(x-4\right)^2-16}\over{x}}}$.

    −8 8 −4 0 4 Does not exist.

  3. Find $\displaystyle \lim_{x\rightarrow -2}{{{x^2-x-6}\over{x^2-2\,x-8}}}$.

    $\displaystyle {{5}\over{6}}$ 5 −5 $\displaystyle -{{5}\over{6}}$ Does not exist.

  4. Find $\displaystyle \lim_{x\rightarrow 2}{{{x-2}\over{\left\vert x-2\right\vert }}}$.

    1 Does not exist. 0 -1

  5. Find $\displaystyle \lim_{x\rightarrow 0}{{{2-\sqrt{x+4}}\over{x\,\sqrt{x+9}}}}$.

    $\displaystyle -{{1}\over{3}}$. $\displaystyle {{1}\over{3}}$. Does not exist. $\displaystyle {{1}\over{12}}$. $\displaystyle -{{1}\over{12}}$.

  6. Find $\displaystyle \lim_{x\rightarrow -4}{\left\vert x+4\right\vert }$.

    $1$ Does not exist. $-1$ 0 4

  7. Find $\displaystyle\lim_{x\to 0} x^2 \cos\left(\frac{1}{x}\right)$

    $-1$ 0 Does not exist. $1$

  8. Find $\displaystyle \lim_{x\rightarrow 2}{{{{{1}\over{x^2}}-{{1}\over{4}}}\over{x^2-4}} }$.

    Does not exist. $\displaystyle -{{1}\over{4}}$ $\displaystyle -{{1}\over{16}}$. $\displaystyle {{1}\over{16}}$. $\displaystyle {{1}\over{4}}$

  9. If −1 $\le f(x) \le$ 1 for all $x$, find $\displaystyle \lim_{x\rightarrow 0}{x^2\,f\left(x\right)}$.

    Does not exist. 1. 0

  10. Find $\displaystyle \lim_{x\rightarrow 4}{{{x^2-2\,x-8}\over{x-4}}}$.

    4. 6. Does not exist. −6. 0


Department of Mathematics
Last modified: 2025-12-21