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

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

  2. If −2 $\le f(x) \le$ $\displaystyle x^2-6\,x+7$ for all $x$, find $\displaystyle \lim_{x\rightarrow 3}{f\left(x\right)}$.

    −2. Does not exist. 0 3.

  3. Find $\displaystyle\lim_{x\to 0} x^3 \cos\left(\frac{1}{x}\right)$

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

  4. Let $\displaystyle f(x) = \begin{cases} x^2-8\,x+15 & \mbox{ if } x \le 4 ; \\ 9-2\,x & \mbox{ if } x > 4 . \end{cases} $

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

    0 1 Does not exist -1 1

  5. Find $\displaystyle \lim_{x\rightarrow -3}{{{x^2+5\,x+6}\over{x+3}}}$.

    −3. 1. Does not exist. −1. 0

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

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

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

    0 8 −8 Does not exist. 4 −4

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

    $\displaystyle -{{1}\over{4}}$. $\displaystyle -{{1}\over{2}}$ $\displaystyle {{1}\over{4}}$. Does not exist. $\displaystyle {{1}\over{2}}$

  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 $\displaystyle \lim_{x\rightarrow 0}{{{\left(x-1\right)^2-1}\over{x}}}$.

    2 −1 1 Does not exist. 0 −2


Department of Mathematics
Last modified: 2026-07-15