1. Find $\displaystyle \lim_{x\rightarrow -3}{{{x^2+2\,x-3}\over{x^2-x-12}}}$.

    $\displaystyle -{{4}\over{7}}$ −4 4 $\displaystyle {{4}\over{7}}$ Does not exist.

  2. Let $\displaystyle f(x) = \begin{cases} x^2-8\,x+14 & \mbox{ if } x \le 4 ; \\ 9-2\,x & \mbox{ if } x > 4 . \end{cases} $

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

    -2 1 Does not exist 0 2

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

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

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

    4 Does not exist. −4 −2 0 2

  5. Find $\displaystyle \lim_{x\rightarrow 4}{{{x^2-3\,x-4}\over{x-4}}}$.

    0 −5. 4. 5. Does not exist.

  6. Find $\displaystyle \lim_{x\rightarrow 0}{{{1-\sqrt{x+1}}\over{x\,\sqrt{x+1}}}}$.

    $\displaystyle {{1}\over{2}}$. Does not exist. 1. −1. $\displaystyle -{{1}\over{2}}$.

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

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

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

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

  9. Find $\displaystyle \lim_{x\rightarrow 16}{{{16-x}\over{4-\sqrt{x}}}}$.

    −8 Does not exist. 4 8 0 −4

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

    4. 0 Does not exist.


Department of Mathematics
Last modified: 2026-05-18