AP Calculus AB Flashcards: Connecting Infinite Limits And Vertical Asymptotes

Study Connecting Infinite Limits And Vertical Asymptotes in AP Calculus AB with focused flashcards that help you recognize the idea, recall the key rule, and apply it in practice-style prompts.

AP Calculus AB

Connecting Infinite Limits And Vertical Asymptotes

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QUESTION

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

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ANSWER

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

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What this deck covers

This deck focuses on Connecting Infinite Limits And Vertical Asymptotes, giving you a quick way to review the definitions, rules, and examples that matter most for AP Calculus AB.

How to use these flashcards

Work through these flashcards in short sessions. Try to answer each prompt before flipping the card, then revisit any cards you miss until the explanation feels automatic.

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

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

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

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

Flashcard 3: Find the vertical asymptotes in $g(x) = \frac{x^2}{x^2 + 4x + 4}$ .

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

Flashcard 4: Identify the vertical asymptote in $f(x) = \frac{1}{x-2}$ .

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

Flashcard 5: Identify the asymptotes of $f(x) = \frac{2x}{x^2 - 4x + 4}$ .

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

Flashcard 6: Identify the asymptote of $f(x) = \frac{2}{x+3}$ .

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

Flashcard 7: Identify the asymptote of $f(x) = \frac{2}{x+3}$ .

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

Flashcard 8: What is the condition for a vertical asymptote at $x = a$ in $f(x) = \frac{p(x)}{q(x)}$ ?

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

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

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

Flashcard 10: What does $\text{lim}_{x \to a^-} f(x) = -\text{infinity}$ indicate?

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

Flashcard 11: Find the vertical asymptotes of $f(x) = \frac{x}{x^2 - 9}$ .

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

Flashcard 12: Determine the vertical asymptote in $g(x) = \frac{x^3}{x^2 - x - 6}$ .

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

Flashcard 13: Find the vertical asymptotes in $g(x) = \frac{x^2}{x^2 + 4x + 4}$ .

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

Flashcard 14: What does $\text{lim}_{x \to a^+} f(x) = \text{infinity}$ tell about $f(x)$ ?

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

Flashcard 15: Find the vertical asymptote of $f(x) = \frac{x}{x^2 - x - 2}$ .

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

Flashcard 16: What is the condition for a vertical asymptote at $x = a$ in $f(x) = \frac{p(x)}{q(x)}$ ?

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

Flashcard 17: What is the behavior of $f(x)$ near a vertical asymptote at $x = a$ ?

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

Flashcard 18: Identify the asymptotes of $f(x) = \frac{1}{x^2 - 4x + 3}$ .

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

Flashcard 19: What does $\text{lim}_{x \to a} f(x) = -\text{infinity}$ imply about $f(x)$ ?

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

Flashcard 20: What does $\text{lim}_{x \to a} f(x) = \text{infinity}$ imply about $f(x)$ ?

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

Flashcard 21: What does $\text{lim}_{x \to a} f(x) = \text{infinity}$ imply about $f(x)$ ?

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

Flashcard 22: What does $\text{lim}_{x \to a^+} f(x) = \text{infinity}$ tell about $f(x)$ ?

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

Flashcard 23: What does $\text{lim}_{x \to a^+} f(x) = -\text{infinity}$ indicate?

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

Flashcard 24: What is the behavior of $f(x)$ near a vertical asymptote?

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

Flashcard 25: Find the vertical asymptotes of $f(x) = \frac{x}{x^2 - 16}$ .

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

Flashcard 26: Determine the vertical asymptote of $f(x) = \frac{1}{(x-1)(x+2)}$ .

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

Flashcard 27: State the condition for a vertical asymptote in $f(x) = \frac{p(x)}{q(x)}$ .

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

Flashcard 28: Find the vertical asymptote of $g(x) = \frac{1}{x^2 - 1}$ .

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

Flashcard 29: Find the vertical asymptotes of $f(x) = \frac{3x}{x^2 + 2x - 3}$ .

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

Flashcard 30: What is the behavior of $f(x)$ if $\text{lim}_{x \to a^-} f(x) = \text{infinity}$ ?

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

Flashcard 31: Calculate the vertical asymptotes of $f(x) = \frac{x^2 + 1}{x - 5}$ .

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

Flashcard 32: Find the vertical asymptotes of $f(x) = \frac{x}{x^2 - 16}$ .

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

Flashcard 33: Identify the vertical asymptotes of $f(x) = \frac{2x}{x^2 - 4}$ .

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

Flashcard 34: Determine the asymptote of $f(x) = \frac{3x}{x^2 - 2x}$ .

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

Flashcard 35: Determine the asymptote of $f(x) = \frac{3x}{x^2 - 2x}$ .

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

Flashcard 36: What does it mean if $\text{lim}_{x \to a^-} f(x) = \text{infinity}$ ?

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

Flashcard 37: What does it mean if $\text{lim}_{x \to a^-} f(x) = \text{infinity}$ ?

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

Flashcard 38: Identify the vertical asymptotes of $f(x) = \frac{2}{x^2 - 9x + 18}$ .

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

Flashcard 39: What does $\text{lim}_{x \to a} f(x) = -\text{infinity}$ imply about $f(x)$ ?

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

Flashcard 40: What does $\text{lim}_{x \to a^+} f(x) = -\text{infinity}$ indicate?

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

Flashcard 41: Identify the vertical asymptotes of $f(x) = \frac{x^2}{x^2 - 4}$ .

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

Flashcard 42: Identify the vertical asymptotes of $f(x) = \frac{2x}{x^2 - 4}$ .

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

Flashcard 43: State the condition for a vertical asymptote in $f(x) = \frac{p(x)}{q(x)}$ .

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

Flashcard 44: Calculate the vertical asymptotes of $f(x) = \frac{x^2 + 1}{x - 5}$ .

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

Flashcard 45: What is the behavior of $f(x)$ near a vertical asymptote at $x = a$ ?

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

Flashcard 46: Identify the vertical asymptote in $f(x) = \frac{1}{x-2}$ .

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

Flashcard 47: What does $\text{lim}_{x \to a^+} f(x) = -\text{infinity}$ tell us?

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

Flashcard 48: Find the vertical asymptotes of $f(x) = \frac{3x}{x^2 + 2x - 3}$ .

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

Flashcard 49: Identify the vertical asymptotes of $f(x) = \frac{x^2}{x^2 - 4}$ .

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

Flashcard 50: What does $\text{lim}_{x \to a^+} f(x) = -\text{infinity}$ tell us?

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

Flashcard 51: What does $\text{lim}_{x \to a^-} f(x) = -\text{infinity}$ indicate?

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

Flashcard 52: Determine the vertical asymptotes of $f(x) = \frac{x^2 + 2}{x^2 - 4x + 4}$ .

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

Flashcard 53: What is the behavior of $f(x)$ near a vertical asymptote?

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

Flashcard 54: Determine the vertical asymptote of $f(x) = \frac{1}{(x-1)(x+2)}$ .

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

Flashcard 55: What does $\text{lim}_{x \to a} f(x) = \text{infinity}$ imply for $f(x)$ ?

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

Flashcard 56: Determine the vertical asymptotes of $f(x) = \frac{x^2 + 2}{x^2 - 4x + 4}$ .

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

Flashcard 57: Identify the asymptotes of $f(x) = \frac{2x}{x^2 - 4x + 4}$ .

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

Flashcard 58: Find the vertical asymptote of $f(x) = \frac{x}{x^2 - x - 2}$ .

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

Flashcard 59: Determine the vertical asymptote in $g(x) = \frac{x^3}{x^2 - x - 6}$ .

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

Flashcard 60: What happens if $\text{lim}_{x \to a^-} f(x) = -\text{infinity}$ ?

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

Flashcard 61: What does it mean if $\text{lim}_{x \to a^+} f(x) = \text{infinity}$ ?

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

Flashcard 62: What is the behavior of $f(x)$ if $\text{lim}_{x \to a^-} f(x) = \text{infinity}$ ?

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

Flashcard 63: Find the vertical asymptote of $g(x) = \frac{1}{x^2 - 1}$ .

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

Flashcard 64: What does $\text{lim}_{x \to a} f(x) = \text{infinity}$ imply for $f(x)$ ?

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

Flashcard 65: What does it mean if $\text{lim}_{x \to a^+} f(x) = \text{infinity}$ ?

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

Flashcard 66: Identify the asymptotes of $f(x) = \frac{1}{x^2 - 4x + 3}$ .

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

AP Calculus AB Flashcards: Connecting Infinite Limits And Vertical Asymptotes

Connecting Infinite Limits And Vertical Asymptotes

What this deck covers

How to use these flashcards

All flashcards

Flashcard 1: Find the vertical asymptotes of f(x)=xx2−9f(x) = \frac{x}{x^2 - 9}f(x)=x2−9x​.

Flashcard 2: What happens if limx→a−f(x)=−infinity\text{lim}_{x \to a^-} f(x) = -\text{infinity}limx→a−​f(x)=−infinity?

Flashcard 3: Find the vertical asymptotes in g(x)=x2x2+4x+4g(x) = \frac{x^2}{x^2 + 4x + 4}g(x)=x2+4x+4x2​.

Flashcard 4: Identify the vertical asymptote in f(x)=1x−2f(x) = \frac{1}{x-2}f(x)=x−21​.

Flashcard 5: Identify the asymptotes of f(x)=2xx2−4x+4f(x) = \frac{2x}{x^2 - 4x + 4}f(x)=x2−4x+42x​.

Flashcard 6: Identify the asymptote of f(x)=2x+3f(x) = \frac{2}{x+3}f(x)=x+32​.

Flashcard 7: Identify the asymptote of f(x)=2x+3f(x) = \frac{2}{x+3}f(x)=x+32​.

Flashcard 8: What is the condition for a vertical asymptote at x=ax = ax=a in f(x)=p(x)q(x)f(x) = \frac{p(x)}{q(x)}f(x)=q(x)p(x)​?

Flashcard 9: Identify the vertical asymptotes of f(x)=2x2−9x+18f(x) = \frac{2}{x^2 - 9x + 18}f(x)=x2−9x+182​.

Flashcard 10: What does limx→a−f(x)=−infinity\text{lim}_{x \to a^-} f(x) = -\text{infinity}limx→a−​f(x)=−infinity indicate?

Flashcard 11: Find the vertical asymptotes of f(x)=xx2−9f(x) = \frac{x}{x^2 - 9}f(x)=x2−9x​.

Flashcard 12: Determine the vertical asymptote in g(x)=x3x2−x−6g(x) = \frac{x^3}{x^2 - x - 6}g(x)=x2−x−6x3​.

Flashcard 13: Find the vertical asymptotes in g(x)=x2x2+4x+4g(x) = \frac{x^2}{x^2 + 4x + 4}g(x)=x2+4x+4x2​.

Flashcard 14: What does limx→a+f(x)=infinity\text{lim}_{x \to a^+} f(x) = \text{infinity}limx→a+​f(x)=infinity tell about f(x)f(x)f(x)?

Flashcard 15: Find the vertical asymptote of f(x)=xx2−x−2f(x) = \frac{x}{x^2 - x - 2}f(x)=x2−x−2x​.

Flashcard 16: What is the condition for a vertical asymptote at x=ax = ax=a in f(x)=p(x)q(x)f(x) = \frac{p(x)}{q(x)}f(x)=q(x)p(x)​?

Flashcard 17: What is the behavior of f(x)f(x)f(x) near a vertical asymptote at x=ax = ax=a?

Flashcard 18: Identify the asymptotes of f(x)=1x2−4x+3f(x) = \frac{1}{x^2 - 4x + 3}f(x)=x2−4x+31​.

Flashcard 19: What does limx→af(x)=−infinity\text{lim}_{x \to a} f(x) = -\text{infinity}limx→a​f(x)=−infinity imply about f(x)f(x)f(x)?

Flashcard 20: What does limx→af(x)=infinity\text{lim}_{x \to a} f(x) = \text{infinity}limx→a​f(x)=infinity imply about f(x)f(x)f(x)?

Flashcard 21: What does limx→af(x)=infinity\text{lim}_{x \to a} f(x) = \text{infinity}limx→a​f(x)=infinity imply about f(x)f(x)f(x)?

Flashcard 22: What does limx→a+f(x)=infinity\text{lim}_{x \to a^+} f(x) = \text{infinity}limx→a+​f(x)=infinity tell about f(x)f(x)f(x)?

Flashcard 23: What does limx→a+f(x)=−infinity\text{lim}_{x \to a^+} f(x) = -\text{infinity}limx→a+​f(x)=−infinity indicate?

Flashcard 24: What is the behavior of f(x)f(x)f(x) near a vertical asymptote?

Flashcard 25: Find the vertical asymptotes of f(x)=xx2−16f(x) = \frac{x}{x^2 - 16}f(x)=x2−16x​.

Flashcard 26: Determine the vertical asymptote of f(x)=1(x−1)(x+2)f(x) = \frac{1}{(x-1)(x+2)}f(x)=(x−1)(x+2)1​.

Flashcard 27: State the condition for a vertical asymptote in f(x)=p(x)q(x)f(x) = \frac{p(x)}{q(x)}f(x)=q(x)p(x)​.

Flashcard 28: Find the vertical asymptote of g(x)=1x2−1g(x) = \frac{1}{x^2 - 1}g(x)=x2−11​.

Flashcard 29: Find the vertical asymptotes of f(x)=3xx2+2x−3f(x) = \frac{3x}{x^2 + 2x - 3}f(x)=x2+2x−33x​.

Flashcard 30: What is the behavior of f(x)f(x)f(x) if limx→a−f(x)=infinity\text{lim}_{x \to a^-} f(x) = \text{infinity}limx→a−​f(x)=infinity?

Flashcard 31: Calculate the vertical asymptotes of f(x)=x2+1x−5f(x) = \frac{x^2 + 1}{x - 5}f(x)=x−5x2+1​.

Flashcard 32: Find the vertical asymptotes of f(x)=xx2−16f(x) = \frac{x}{x^2 - 16}f(x)=x2−16x​.

Flashcard 33: Identify the vertical asymptotes of f(x)=2xx2−4f(x) = \frac{2x}{x^2 - 4}f(x)=x2−42x​.

Flashcard 34: Determine the asymptote of f(x)=3xx2−2xf(x) = \frac{3x}{x^2 - 2x}f(x)=x2−2x3x​.

Flashcard 35: Determine the asymptote of f(x)=3xx2−2xf(x) = \frac{3x}{x^2 - 2x}f(x)=x2−2x3x​.

Flashcard 36: What does it mean if limx→a−f(x)=infinity\text{lim}_{x \to a^-} f(x) = \text{infinity}limx→a−​f(x)=infinity?

Flashcard 37: What does it mean if limx→a−f(x)=infinity\text{lim}_{x \to a^-} f(x) = \text{infinity}limx→a−​f(x)=infinity?

Flashcard 38: Identify the vertical asymptotes of f(x)=2x2−9x+18f(x) = \frac{2}{x^2 - 9x + 18}f(x)=x2−9x+182​.

Flashcard 39: What does limx→af(x)=−infinity\text{lim}_{x \to a} f(x) = -\text{infinity}limx→a​f(x)=−infinity imply about f(x)f(x)f(x)?

Flashcard 40: What does limx→a+f(x)=−infinity\text{lim}_{x \to a^+} f(x) = -\text{infinity}limx→a+​f(x)=−infinity indicate?

Flashcard 41: Identify the vertical asymptotes of f(x)=x2x2−4f(x) = \frac{x^2}{x^2 - 4}f(x)=x2−4x2​.

Flashcard 42: Identify the vertical asymptotes of f(x)=2xx2−4f(x) = \frac{2x}{x^2 - 4}f(x)=x2−42x​.

Flashcard 43: State the condition for a vertical asymptote in f(x)=p(x)q(x)f(x) = \frac{p(x)}{q(x)}f(x)=q(x)p(x)​.

Flashcard 44: Calculate the vertical asymptotes of f(x)=x2+1x−5f(x) = \frac{x^2 + 1}{x - 5}f(x)=x−5x2+1​.

Flashcard 45: What is the behavior of f(x)f(x)f(x) near a vertical asymptote at x=ax = ax=a?

Flashcard 46: Identify the vertical asymptote in f(x)=1x−2f(x) = \frac{1}{x-2}f(x)=x−21​.

Flashcard 47: What does limx→a+f(x)=−infinity\text{lim}_{x \to a^+} f(x) = -\text{infinity}limx→a+​f(x)=−infinity tell us?

Flashcard 48: Find the vertical asymptotes of f(x)=3xx2+2x−3f(x) = \frac{3x}{x^2 + 2x - 3}f(x)=x2+2x−33x​.

Flashcard 49: Identify the vertical asymptotes of f(x)=x2x2−4f(x) = \frac{x^2}{x^2 - 4}f(x)=x2−4x2​.

Flashcard 50: What does limx→a+f(x)=−infinity\text{lim}_{x \to a^+} f(x) = -\text{infinity}limx→a+​f(x)=−infinity tell us?

Flashcard 51: What does limx→a−f(x)=−infinity\text{lim}_{x \to a^-} f(x) = -\text{infinity}limx→a−​f(x)=−infinity indicate?

Flashcard 52: Determine the vertical asymptotes of f(x)=x2+2x2−4x+4f(x) = \frac{x^2 + 2}{x^2 - 4x + 4}f(x)=x2−4x+4x2+2​.

Flashcard 53: What is the behavior of f(x)f(x)f(x) near a vertical asymptote?

Flashcard 54: Determine the vertical asymptote of f(x)=1(x−1)(x+2)f(x) = \frac{1}{(x-1)(x+2)}f(x)=(x−1)(x+2)1​.

Flashcard 55: What does limx→af(x)=infinity\text{lim}_{x \to a} f(x) = \text{infinity}limx→a​f(x)=infinity imply for f(x)f(x)f(x)?

Flashcard 56: Determine the vertical asymptotes of f(x)=x2+2x2−4x+4f(x) = \frac{x^2 + 2}{x^2 - 4x + 4}f(x)=x2−4x+4x2+2​.

Flashcard 57: Identify the asymptotes of f(x)=2xx2−4x+4f(x) = \frac{2x}{x^2 - 4x + 4}f(x)=x2−4x+42x​.

Flashcard 58: Find the vertical asymptote of f(x)=xx2−x−2f(x) = \frac{x}{x^2 - x - 2}f(x)=x2−x−2x​.

Flashcard 59: Determine the vertical asymptote in g(x)=x3x2−x−6g(x) = \frac{x^3}{x^2 - x - 6}g(x)=x2−x−6x3​.

Flashcard 60: What happens if limx→a−f(x)=−infinity\text{lim}_{x \to a^-} f(x) = -\text{infinity}limx→a−​f(x)=−infinity?

Flashcard 61: What does it mean if limx→a+f(x)=infinity\text{lim}_{x \to a^+} f(x) = \text{infinity}limx→a+​f(x)=infinity?

Flashcard 62: What is the behavior of f(x)f(x)f(x) if limx→a−f(x)=infinity\text{lim}_{x \to a^-} f(x) = \text{infinity}limx→a−​f(x)=infinity?

Flashcard 63: Find the vertical asymptote of g(x)=1x2−1g(x) = \frac{1}{x^2 - 1}g(x)=x2−11​.

Flashcard 64: What does limx→af(x)=infinity\text{lim}_{x \to a} f(x) = \text{infinity}limx→a​f(x)=infinity imply for f(x)f(x)f(x)?

Flashcard 65: What does it mean if limx→a+f(x)=infinity\text{lim}_{x \to a^+} f(x) = \text{infinity}limx→a+​f(x)=infinity?

Flashcard 66: Identify the asymptotes of f(x)=1x2−4x+3f(x) = \frac{1}{x^2 - 4x + 3}f(x)=x2−4x+31​.

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

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

Flashcard 3: Find the vertical asymptotes in $g(x) = \frac{x^2}{x^2 + 4x + 4}$ .

Flashcard 4: Identify the vertical asymptote in $f(x) = \frac{1}{x-2}$ .

Flashcard 5: Identify the asymptotes of $f(x) = \frac{2x}{x^2 - 4x + 4}$ .

Flashcard 6: Identify the asymptote of $f(x) = \frac{2}{x+3}$ .

Flashcard 7: Identify the asymptote of $f(x) = \frac{2}{x+3}$ .

Flashcard 8: What is the condition for a vertical asymptote at $x = a$ in $f(x) = \frac{p(x)}{q(x)}$ ?

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

Flashcard 10: What does $\text{lim}_{x \to a^-} f(x) = -\text{infinity}$ indicate?

Flashcard 11: Find the vertical asymptotes of $f(x) = \frac{x}{x^2 - 9}$ .

Flashcard 12: Determine the vertical asymptote in $g(x) = \frac{x^3}{x^2 - x - 6}$ .

Flashcard 13: Find the vertical asymptotes in $g(x) = \frac{x^2}{x^2 + 4x + 4}$ .

Flashcard 14: What does $\text{lim}_{x \to a^+} f(x) = \text{infinity}$ tell about $f(x)$ ?

Flashcard 15: Find the vertical asymptote of $f(x) = \frac{x}{x^2 - x - 2}$ .

Flashcard 16: What is the condition for a vertical asymptote at $x = a$ in $f(x) = \frac{p(x)}{q(x)}$ ?

Flashcard 17: What is the behavior of $f(x)$ near a vertical asymptote at $x = a$ ?

Flashcard 18: Identify the asymptotes of $f(x) = \frac{1}{x^2 - 4x + 3}$ .

Flashcard 19: What does $\text{lim}_{x \to a} f(x) = -\text{infinity}$ imply about $f(x)$ ?

Flashcard 20: What does $\text{lim}_{x \to a} f(x) = \text{infinity}$ imply about $f(x)$ ?

Flashcard 21: What does $\text{lim}_{x \to a} f(x) = \text{infinity}$ imply about $f(x)$ ?

Flashcard 22: What does $\text{lim}_{x \to a^+} f(x) = \text{infinity}$ tell about $f(x)$ ?

Flashcard 23: What does $\text{lim}_{x \to a^+} f(x) = -\text{infinity}$ indicate?

Flashcard 24: What is the behavior of $f(x)$ near a vertical asymptote?

Flashcard 25: Find the vertical asymptotes of $f(x) = \frac{x}{x^2 - 16}$ .

Flashcard 26: Determine the vertical asymptote of $f(x) = \frac{1}{(x-1)(x+2)}$ .

Flashcard 27: State the condition for a vertical asymptote in $f(x) = \frac{p(x)}{q(x)}$ .

Flashcard 28: Find the vertical asymptote of $g(x) = \frac{1}{x^2 - 1}$ .

Flashcard 29: Find the vertical asymptotes of $f(x) = \frac{3x}{x^2 + 2x - 3}$ .

Flashcard 30: What is the behavior of $f(x)$ if $\text{lim}_{x \to a^-} f(x) = \text{infinity}$ ?

Flashcard 31: Calculate the vertical asymptotes of $f(x) = \frac{x^2 + 1}{x - 5}$ .

Flashcard 32: Find the vertical asymptotes of $f(x) = \frac{x}{x^2 - 16}$ .

Flashcard 33: Identify the vertical asymptotes of $f(x) = \frac{2x}{x^2 - 4}$ .

Flashcard 34: Determine the asymptote of $f(x) = \frac{3x}{x^2 - 2x}$ .

Flashcard 35: Determine the asymptote of $f(x) = \frac{3x}{x^2 - 2x}$ .

Flashcard 36: What does it mean if $\text{lim}_{x \to a^-} f(x) = \text{infinity}$ ?

Flashcard 37: What does it mean if $\text{lim}_{x \to a^-} f(x) = \text{infinity}$ ?

Flashcard 38: Identify the vertical asymptotes of $f(x) = \frac{2}{x^2 - 9x + 18}$ .

Flashcard 39: What does $\text{lim}_{x \to a} f(x) = -\text{infinity}$ imply about $f(x)$ ?

Flashcard 40: What does $\text{lim}_{x \to a^+} f(x) = -\text{infinity}$ indicate?

Flashcard 41: Identify the vertical asymptotes of $f(x) = \frac{x^2}{x^2 - 4}$ .

Flashcard 42: Identify the vertical asymptotes of $f(x) = \frac{2x}{x^2 - 4}$ .

Flashcard 43: State the condition for a vertical asymptote in $f(x) = \frac{p(x)}{q(x)}$ .

Flashcard 44: Calculate the vertical asymptotes of $f(x) = \frac{x^2 + 1}{x - 5}$ .

Flashcard 45: What is the behavior of $f(x)$ near a vertical asymptote at $x = a$ ?

Flashcard 46: Identify the vertical asymptote in $f(x) = \frac{1}{x-2}$ .

Flashcard 47: What does $\text{lim}_{x \to a^+} f(x) = -\text{infinity}$ tell us?

Flashcard 48: Find the vertical asymptotes of $f(x) = \frac{3x}{x^2 + 2x - 3}$ .

Flashcard 49: Identify the vertical asymptotes of $f(x) = \frac{x^2}{x^2 - 4}$ .

Flashcard 50: What does $\text{lim}_{x \to a^+} f(x) = -\text{infinity}$ tell us?

Flashcard 51: What does $\text{lim}_{x \to a^-} f(x) = -\text{infinity}$ indicate?

Flashcard 52: Determine the vertical asymptotes of $f(x) = \frac{x^2 + 2}{x^2 - 4x + 4}$ .

Flashcard 53: What is the behavior of $f(x)$ near a vertical asymptote?

Flashcard 54: Determine the vertical asymptote of $f(x) = \frac{1}{(x-1)(x+2)}$ .

Flashcard 55: What does $\text{lim}_{x \to a} f(x) = \text{infinity}$ imply for $f(x)$ ?

Flashcard 56: Determine the vertical asymptotes of $f(x) = \frac{x^2 + 2}{x^2 - 4x + 4}$ .

Flashcard 57: Identify the asymptotes of $f(x) = \frac{2x}{x^2 - 4x + 4}$ .

Flashcard 58: Find the vertical asymptote of $f(x) = \frac{x}{x^2 - x - 2}$ .

Flashcard 59: Determine the vertical asymptote in $g(x) = \frac{x^3}{x^2 - x - 6}$ .

Flashcard 60: What happens if $\text{lim}_{x \to a^-} f(x) = -\text{infinity}$ ?

Flashcard 61: What does it mean if $\text{lim}_{x \to a^+} f(x) = \text{infinity}$ ?

Flashcard 62: What is the behavior of $f(x)$ if $\text{lim}_{x \to a^-} f(x) = \text{infinity}$ ?

Flashcard 63: Find the vertical asymptote of $g(x) = \frac{1}{x^2 - 1}$ .

Flashcard 64: What does $\text{lim}_{x \to a} f(x) = \text{infinity}$ imply for $f(x)$ ?

Flashcard 65: What does it mean if $\text{lim}_{x \to a^+} f(x) = \text{infinity}$ ?

Flashcard 66: Identify the asymptotes of $f(x) = \frac{1}{x^2 - 4x + 3}$ .