Connecting Infinite Limits and Vertical Asymptotes - AP Calculus AB

1. Find the vertical asymptotes of $f(x) = \frac{x}{x^2 - 9}$.

   Vertical asymptotes at $x = 3$ and $x = -3$. Factor denominator: $x^2 - 9 = (x-3)(x+3) = 0$.

2. What happens if $\text{lim}_{x \to a^-} f(x) = -\text{infinity}$?

   $f(x)$ has a vertical asymptote at $x = a$. Left-sided negative limit confirms vertical asymptote.

3. Identify the vertical asymptote in $f(x) = \frac{1}{x-2}$.

   Vertical asymptote at $x = 2$. Denominator equals zero when $x - 2 = 0$.

4. Identify the asymptote of $f(x) = \frac{2}{x+3}$.

   Vertical asymptote at $x = -3$. Denominator equals zero when $x + 3 = 0$.

5. What is the condition for a vertical asymptote at $x = a$ in $f(x) = \frac{p(x)}{q(x)}$?

   $q(a) = 0$ and $p(a) \neq 0$. Denominator zero with non-zero numerator creates asymptote.

6. Identify the vertical asymptotes of $f(x) = \frac{2}{x^2 - 9x + 18}$.

   Vertical asymptotes at $x = 3$ and $x = 6$. Factor $x^2 - 9x + 18 = (x-3)(x-6) = 0$.

7. Determine the vertical asymptote in $g(x) = \frac{x^3}{x^2 - x - 6}$.

   Vertical asymptotes at $x = 3$ and $x = -2$. Factor $x^2 - x - 6 = (x-3)(x+2) = 0$.

8. Find the vertical asymptotes in $g(x) = \frac{x^2}{x^2 + 4x + 4}$.

   Vertical asymptote at $x = -2$. Denominator $(x+2)^2 = 0$ only when $x = -2$.

9. What does $\text{lim}_{x \to a^+} f(x) = \text{infinity}$ tell about $f(x)$?

   $f(x)$ approaches infinity at $x = a$ from the right. Describes function behavior approaching from the right.

10. Find the vertical asymptote of $f(x) = \frac{x}{x^2 - x - 2}$.

    Vertical asymptotes at $x = 2$ and $x = -1$. Factor $x^2 - x - 2 = (x-2)(x+1) = 0$.

11. What does $\text{lim}_{x \to a} f(x) = \text{infinity}$ imply about $f(x)$?

    $f(x)$ has a vertical asymptote at $x = a$. An infinite limit indicates unbounded growth near that point.

12. What does $\text{lim}_{x \to a^+} f(x) = -\text{infinity}$ indicate?

    $f(x)$ has a vertical asymptote at $x = a$. Right-sided negative infinite limit confirms asymptote.

13. Find the vertical asymptotes of $f(x) = \frac{x}{x^2 - 16}$.

    Vertical asymptotes at $x = 4$ and $x = -4$. Factor denominator: $x^2 - 16 = (x-4)(x+4) = 0$.

14. Determine the vertical asymptote of $f(x) = \frac{1}{(x-1)(x+2)}$.

    Vertical asymptotes at $x = 1$ and $x = -2$. Each factor in denominator creates a separate asymptote.

15. State the condition for a vertical asymptote in $f(x) = \frac{p(x)}{q(x)}$.

    $q(x) = 0$ at $x = a$ and $p(x) \neq 0$. Standard condition for rational function vertical asymptotes.

16. Find the vertical asymptote of $g(x) = \frac{1}{x^2 - 1}$.

    Vertical asymptotes at $x = 1$ and $x = -1$. Factor denominator: $x^2 - 1 = (x-1)(x+1) = 0$.

17. Find the vertical asymptotes of $f(x) = \frac{3x}{x^2 + 2x - 3}$.

    Vertical asymptotes at $x = 1$ and $x = -3$. Factor $x^2 + 2x - 3 = (x-1)(x+3) = 0$.

18. What is the behavior of $f(x)$ if $\text{lim}_{x \to a^-} f(x) = \text{infinity}$?

    $f(x)$ approaches infinity at $x = a$ from the left. Left-sided limit describes approach from negative direction.

19. Identify the vertical asymptotes of $f(x) = \frac{2x}{x^2 - 4}$.

    Vertical asymptotes at $x = 2$ and $x = -2$. Factor denominator: $x^2 - 4 = (x-2)(x+2) = 0$.

20. Determine the asymptote of $f(x) = \frac{3x}{x^2 - 2x}$.

    Vertical asymptote at $x = 0$ and $x = 2$. Factor out $x$: denominator $x(x-2) = 0$.

21. What does it mean if $\text{lim}_{x \to a^-} f(x) = \text{infinity}$?

    $f(x)$ has a vertical asymptote at $x = a$. Left-sided infinite limit confirms vertical asymptote exists.

22. What does $\text{lim}_{x \to a} f(x) = -\text{infinity}$ imply about $f(x)$?

    $f(x)$ has a vertical asymptote at $x = a$. Negative infinite limit still indicates vertical asymptote.

23. Calculate the vertical asymptotes of $f(x) = \frac{x^2 + 1}{x - 5}$.

    Vertical asymptote at $x = 5$. Denominator equals zero only when $x - 5 = 0$.

24. What is the behavior of $f(x)$ near a vertical asymptote at $x = a$?

    $f(x)$ approaches infinity or negative infinity. Function values become unbounded near asymptotes.

25. Identify the vertical asymptotes of $f(x) = \frac{x^2}{x^2 - 4}$.

    Vertical asymptotes at $x = 2$ and $x = -2$. Factor denominator: $(x-2)(x+2) = 0$ gives both roots.

26. What does $\text{lim}_{x \to a^+} f(x) = -\text{infinity}$ tell us?

    $f(x)$ has a vertical asymptote at $x = a$. Right-sided negative limit confirms vertical asymptote.

27. What does $\text{lim}_{x \to a^-} f(x) = -\text{infinity}$ indicate?

    $f(x)$ has a vertical asymptote at $x = a$. Left-sided infinite limit confirms vertical asymptote exists.

28. Determine the vertical asymptotes of $f(x) = \frac{x^2 + 2}{x^2 - 4x + 4}$.

    Vertical asymptote at $x = 2$. Denominator $(x-2)^2 = 0$ only when $x = 2$.

29. What is the behavior of $f(x)$ near a vertical asymptote?

    $f(x)$ approaches infinity or negative infinity. Function values grow without bound near the asymptote.

30. Identify the asymptotes of $f(x) = \frac{2x}{x^2 - 4x + 4}$.

    Vertical asymptote at $x = 2$. Denominator $x^2 - 4x + 4 = (x-2)^2 = 0$ at $x = 2$.

31. What does $\text{lim}_{x \to a} f(x) = \text{infinity}$ imply for $f(x)$?

    $f(x)$ has a vertical asymptote at $x = a$. Two-sided infinite limit confirms vertical asymptote.

32. What does it mean if $\text{lim}_{x \to a^+} f(x) = \text{infinity}$?

    $f(x)$ has a vertical asymptote at $x = a$. Right-sided infinite limit confirms vertical asymptote exists.

33. Identify the asymptotes of $f(x) = \frac{1}{x^2 - 4x + 3}$.

    Vertical asymptotes at $x = 1$ and $x = 3$. Factor $x^2 - 4x + 3 = (x-1)(x-3) = 0$.

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