1. Find the vertical asymptotes for $y = \displaystyle {{x-x^2}\over{x^2-1}} $ .

    x = 0 x = 1 x = 1 x = −1 and x = 1 No vertical asymptote x = 1 and x = 0 x = −1

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

    Does not exist. 0 $\displaystyle -\infty $. −2. $\displaystyle \infty $. $\displaystyle {{1}\over{4}}$

  3. Find $\displaystyle \lim_{x\rightarrow \infty }{{{\sqrt{9\,x^4+4\,x}}\over{2\,x-4\,x^2 }}}$

    Does not exist. $\displaystyle {{4}\over{3}}$ $\displaystyle -{{3}\over{4}}$ $\displaystyle -{{4}\over{3}}$ $\displaystyle {{3}\over{4}}$

  4. Find $\displaystyle\lim_{x\to 3 \mbox{ - }} -{{1}\over{x-3}} $ .

    3 Does not exist. $\displaystyle -\infty $ $\displaystyle \infty $ 0

  5. Find $\displaystyle\lim_{x\to \mbox{ } \infty} {{\pi}\over{x-\pi}}$

    $\displaystyle \infty $ $1$ Does not exist. 0

  6. Find $\displaystyle \lim_{x\rightarrow -\infty }{{{x}\over{\sqrt{4\,x^2+4}}}}$ .

    0 $\displaystyle {{1}\over{2}}$ $\displaystyle -{{1}\over{2}}$ Does not exist. $\displaystyle -{{1}\over{2}}$ $\displaystyle {{1}\over{2}}$

  7. Find $\displaystyle\lim_{x\to\infty} \left( \sqrt{9\,x^2+8\,x}-3\,x \right)$ .

    Does not exist. $\displaystyle -{{3}\over{4}}$ 0 $\displaystyle -{{4}\over{3}}$ $\displaystyle {{4}\over{3}}$ $\displaystyle {{3}\over{4}}$

  8. Find $\displaystyle\lim_{x\to 3 \mbox{ + }} {{-x^2-2\,x+8}\over{x-3}} $ .

    3 0 $\displaystyle \infty $ Does not exist. $\displaystyle -\infty $

  9. Find $\displaystyle -\lim_{x\rightarrow 3}{{{1}\over{x-3}}}$.

    $-\infty$ 0 $\infty$ 3 Does not exist.

  10. Find $\displaystyle \lim_{x\rightarrow -{{\pi}\over{2}}}{\tan x}$.

    $\displaystyle \infty $ 0 $\displaystyle -\infty $ $\displaystyle -{{\pi}\over{2}}$ Does not exist.


Department of Mathematics
Last modified: 2025-04-29