1. Find the horizontal asymptotes for $y = \displaystyle {{2\,x^2-9\,x+4}\over{x^2+x-2}} $ .

    y = 2 and y = −2 $\displaystyle y={{1}\over{2}}$ No horizontal asymptote y = −2 y = 4 $\displaystyle y={{1}\over{2}}$ and y = 4 y = 2

  2. Find the vertical asymptotes for $y = \displaystyle {{x^2-5\,x+6}\over{x^2+6\,x+8}} $ .

    x = 2 and x = 3 x = −2 x = −4 x = −2 and x = −4 x = 2 x = 3 No vertical asymptote

  3. Find $\displaystyle \lim_{x\rightarrow 3}{{{1}\over{x^2-6\,x+9}}}$.

    $\displaystyle -\infty $. 0 $\displaystyle \infty $. Does not exist. 3. $\displaystyle {{1}\over{9}}$

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

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

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

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

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

    Does not exist. 5 −5 $\displaystyle {{1}\over{5}}$ $\displaystyle -{{1}\over{5}}$

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

    $\displaystyle -{{2}\over{5}}$ $\displaystyle {{5}\over{2}}$ Does not exist. $\displaystyle -{{5}\over{2}}$ $\displaystyle {{2}\over{5}}$

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

    $\displaystyle {{6}\over{7}}$ $\displaystyle {{7}\over{6}}$ 0 $\displaystyle -{{7}\over{6}}$ $\displaystyle -{{6}\over{7}}$ Does not exist.

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

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

  10. Find $\displaystyle\lim_{x\to -2 \mbox{ - }} -{{1}\over{x+2}} $ .

    $\displaystyle \infty $ 0 Does not exist. −2 $\displaystyle -\infty $


Department of Mathematics
Last modified: 2026-09-06