Connecting Infinite Limits and Vertical Asymptotes - AP Calculus AB
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What does $\text{lim}_{x \to a} f(x) = -\text{infinity}$ imply about $f(x)$?
What does $\text{lim}_{x \to a} f(x) = -\text{infinity}$ imply about $f(x)$?
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$f(x)$ has a vertical asymptote at $x = a$. Negative infinite limit still indicates vertical asymptote.
$f(x)$ has a vertical asymptote at $x = a$. Negative infinite limit still indicates vertical asymptote.
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Identify the asymptote of $f(x) = \frac{2}{x+3}$.
Identify the asymptote of $f(x) = \frac{2}{x+3}$.
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Vertical asymptote at $x = -3$. Denominator equals zero when $x + 3 = 0$.
Vertical asymptote at $x = -3$. Denominator equals zero when $x + 3 = 0$.
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What happens if $\text{lim}_{x \to a^-} f(x) = -\text{infinity}$?
What happens if $\text{lim}_{x \to a^-} f(x) = -\text{infinity}$?
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$f(x)$ has a vertical asymptote at $x = a$. Left-sided negative limit confirms vertical asymptote.
$f(x)$ has a vertical asymptote at $x = a$. Left-sided negative limit confirms vertical asymptote.
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Determine the vertical asymptote in $g(x) = \frac{x^3}{x^2 - x - 6}$.
Determine the vertical asymptote in $g(x) = \frac{x^3}{x^2 - x - 6}$.
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Vertical asymptotes at $x = 3$ and $x = -2$. Factor $x^2 - x - 6 = (x-3)(x+2) = 0$.
Vertical asymptotes at $x = 3$ and $x = -2$. Factor $x^2 - x - 6 = (x-3)(x+2) = 0$.
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What does it mean if $\text{lim}_{x \to a^+} f(x) = \text{infinity}$?
What does it mean if $\text{lim}_{x \to a^+} f(x) = \text{infinity}$?
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$f(x)$ has a vertical asymptote at $x = a$. Right-sided infinite limit confirms vertical asymptote exists.
$f(x)$ has a vertical asymptote at $x = a$. Right-sided infinite limit confirms vertical asymptote exists.
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What does $\text{lim}_{x \to a^+} f(x) = -\text{infinity}$ tell us?
What does $\text{lim}_{x \to a^+} f(x) = -\text{infinity}$ tell us?
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$f(x)$ has a vertical asymptote at $x = a$. Right-sided negative limit confirms vertical asymptote.
$f(x)$ has a vertical asymptote at $x = a$. Right-sided negative limit confirms vertical asymptote.
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Find the vertical asymptotes in $g(x) = \frac{x^2}{x^2 + 4x + 4}$.
Find the vertical asymptotes in $g(x) = \frac{x^2}{x^2 + 4x + 4}$.
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Vertical asymptote at $x = -2$. Denominator $(x+2)^2 = 0$ only when $x = -2$.
Vertical asymptote at $x = -2$. Denominator $(x+2)^2 = 0$ only when $x = -2$.
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What does it mean if $\text{lim}_{x \to a^-} f(x) = \text{infinity}$?
What does it mean if $\text{lim}_{x \to a^-} f(x) = \text{infinity}$?
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$f(x)$ has a vertical asymptote at $x = a$. Left-sided infinite limit confirms vertical asymptote exists.
$f(x)$ has a vertical asymptote at $x = a$. Left-sided infinite limit confirms vertical asymptote exists.
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What does $\text{lim}_{x \to a} f(x) = \text{infinity}$ imply for $f(x)$?
What does $\text{lim}_{x \to a} f(x) = \text{infinity}$ imply for $f(x)$?
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$f(x)$ has a vertical asymptote at $x = a$. Two-sided infinite limit confirms vertical asymptote.
$f(x)$ has a vertical asymptote at $x = a$. Two-sided infinite limit confirms vertical asymptote.
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Determine the vertical asymptote of $f(x) = \frac{1}{(x-1)(x+2)}$.
Determine the vertical asymptote of $f(x) = \frac{1}{(x-1)(x+2)}$.
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Vertical asymptotes at $x = 1$ and $x = -2$. Each factor in denominator creates a separate asymptote.
Vertical asymptotes at $x = 1$ and $x = -2$. Each factor in denominator creates a separate asymptote.
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What does $\text{lim}_{x \to a^+} f(x) = -\text{infinity}$ indicate?
What does $\text{lim}_{x \to a^+} f(x) = -\text{infinity}$ indicate?
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$f(x)$ has a vertical asymptote at $x = a$. Right-sided negative infinite limit confirms asymptote.
$f(x)$ has a vertical asymptote at $x = a$. Right-sided negative infinite limit confirms asymptote.
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Identify the vertical asymptotes of $f(x) = \frac{2}{x^2 - 9x + 18}$.
Identify the vertical asymptotes of $f(x) = \frac{2}{x^2 - 9x + 18}$.
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Vertical asymptotes at $x = 3$ and $x = 6$. Factor $x^2 - 9x + 18 = (x-3)(x-6) = 0$.
Vertical asymptotes at $x = 3$ and $x = 6$. Factor $x^2 - 9x + 18 = (x-3)(x-6) = 0$.
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Find the vertical asymptote of $g(x) = \frac{1}{x^2 - 1}$.
Find the vertical asymptote of $g(x) = \frac{1}{x^2 - 1}$.
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Vertical asymptotes at $x = 1$ and $x = -1$. Factor denominator: $x^2 - 1 = (x-1)(x+1) = 0$.
Vertical asymptotes at $x = 1$ and $x = -1$. Factor denominator: $x^2 - 1 = (x-1)(x+1) = 0$.
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Find the vertical asymptotes of $f(x) = \frac{x}{x^2 - 9}$.
Find the vertical asymptotes of $f(x) = \frac{x}{x^2 - 9}$.
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Vertical asymptotes at $x = 3$ and $x = -3$. Factor denominator: $x^2 - 9 = (x-3)(x+3) = 0$.
Vertical asymptotes at $x = 3$ and $x = -3$. Factor denominator: $x^2 - 9 = (x-3)(x+3) = 0$.
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Find the vertical asymptote of $f(x) = \frac{x}{x^2 - x - 2}$.
Find the vertical asymptote of $f(x) = \frac{x}{x^2 - x - 2}$.
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Vertical asymptotes at $x = 2$ and $x = -1$. Factor $x^2 - x - 2 = (x-2)(x+1) = 0$.
Vertical asymptotes at $x = 2$ and $x = -1$. Factor $x^2 - x - 2 = (x-2)(x+1) = 0$.
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Determine the asymptote of $f(x) = \frac{3x}{x^2 - 2x}$.
Determine the asymptote of $f(x) = \frac{3x}{x^2 - 2x}$.
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Vertical asymptote at $x = 0$ and $x = 2$. Factor out $x$: denominator $x(x-2) = 0$.
Vertical asymptote at $x = 0$ and $x = 2$. Factor out $x$: denominator $x(x-2) = 0$.
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Find the vertical asymptotes of $f(x) = \frac{x}{x^2 - 16}$.
Find the vertical asymptotes of $f(x) = \frac{x}{x^2 - 16}$.
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Vertical asymptotes at $x = 4$ and $x = -4$. Factor denominator: $x^2 - 16 = (x-4)(x+4) = 0$.
Vertical asymptotes at $x = 4$ and $x = -4$. Factor denominator: $x^2 - 16 = (x-4)(x+4) = 0$.
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What does $\text{lim}_{x \to a^+} f(x) = \text{infinity}$ tell about $f(x)$?
What does $\text{lim}_{x \to a^+} f(x) = \text{infinity}$ tell about $f(x)$?
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$f(x)$ approaches infinity at $x = a$ from the right. Describes function behavior approaching from the right.
$f(x)$ approaches infinity at $x = a$ from the right. Describes function behavior approaching from the right.
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Identify the vertical asymptotes of $f(x) = \frac{2x}{x^2 - 4}$.
Identify the vertical asymptotes of $f(x) = \frac{2x}{x^2 - 4}$.
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Vertical asymptotes at $x = 2$ and $x = -2$. Factor denominator: $x^2 - 4 = (x-2)(x+2) = 0$.
Vertical asymptotes at $x = 2$ and $x = -2$. Factor denominator: $x^2 - 4 = (x-2)(x+2) = 0$.
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Identify the asymptotes of $f(x) = \frac{2x}{x^2 - 4x + 4}$.
Identify the asymptotes of $f(x) = \frac{2x}{x^2 - 4x + 4}$.
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Vertical asymptote at $x = 2$. Denominator $x^2 - 4x + 4 = (x-2)^2 = 0$ at $x = 2$.
Vertical asymptote at $x = 2$. Denominator $x^2 - 4x + 4 = (x-2)^2 = 0$ at $x = 2$.
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Find the vertical asymptotes of $f(x) = \frac{3x}{x^2 + 2x - 3}$.
Find the vertical asymptotes of $f(x) = \frac{3x}{x^2 + 2x - 3}$.
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Vertical asymptotes at $x = 1$ and $x = -3$. Factor $x^2 + 2x - 3 = (x-1)(x+3) = 0$.
Vertical asymptotes at $x = 1$ and $x = -3$. Factor $x^2 + 2x - 3 = (x-1)(x+3) = 0$.
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Identify the vertical asymptotes of $f(x) = \frac{x^2}{x^2 - 4}$.
Identify the vertical asymptotes of $f(x) = \frac{x^2}{x^2 - 4}$.
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Vertical asymptotes at $x = 2$ and $x = -2$. Factor denominator: $(x-2)(x+2) = 0$ gives both roots.
Vertical asymptotes at $x = 2$ and $x = -2$. Factor denominator: $(x-2)(x+2) = 0$ gives both roots.
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What is the behavior of $f(x)$ if $\text{lim}_{x \to a^-} f(x) = \text{infinity}$?
What is the behavior of $f(x)$ if $\text{lim}_{x \to a^-} f(x) = \text{infinity}$?
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$f(x)$ approaches infinity at $x = a$ from the left. Left-sided limit describes approach from negative direction.
$f(x)$ approaches infinity at $x = a$ from the left. Left-sided limit describes approach from negative direction.
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What does $\text{lim}_{x \to a} f(x) = \text{infinity}$ imply about $f(x)$?
What does $\text{lim}_{x \to a} f(x) = \text{infinity}$ imply about $f(x)$?
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$f(x)$ has a vertical asymptote at $x = a$. An infinite limit indicates unbounded growth near that point.
$f(x)$ has a vertical asymptote at $x = a$. An infinite limit indicates unbounded growth near that point.
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Determine the vertical asymptotes of $f(x) = \frac{x^2 + 2}{x^2 - 4x + 4}$.
Determine the vertical asymptotes of $f(x) = \frac{x^2 + 2}{x^2 - 4x + 4}$.
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Vertical asymptote at $x = 2$. Denominator $(x-2)^2 = 0$ only when $x = 2$.
Vertical asymptote at $x = 2$. Denominator $(x-2)^2 = 0$ only when $x = 2$.
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Calculate the vertical asymptotes of $f(x) = \frac{x^2 + 1}{x - 5}$.
Calculate the vertical asymptotes of $f(x) = \frac{x^2 + 1}{x - 5}$.
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Vertical asymptote at $x = 5$. Denominator equals zero only when $x - 5 = 0$.
Vertical asymptote at $x = 5$. Denominator equals zero only when $x - 5 = 0$.
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What is the behavior of $f(x)$ near a vertical asymptote?
What is the behavior of $f(x)$ near a vertical asymptote?
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$f(x)$ approaches infinity or negative infinity. Function values grow without bound near the asymptote.
$f(x)$ approaches infinity or negative infinity. Function values grow without bound near the asymptote.
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What is the condition for a vertical asymptote at $x = a$ in $f(x) = \frac{p(x)}{q(x)}$?
What is the condition for a vertical asymptote at $x = a$ in $f(x) = \frac{p(x)}{q(x)}$?
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$q(a) = 0$ and $p(a) \neq 0$. Denominator zero with non-zero numerator creates asymptote.
$q(a) = 0$ and $p(a) \neq 0$. Denominator zero with non-zero numerator creates asymptote.
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Identify the asymptotes of $f(x) = \frac{1}{x^2 - 4x + 3}$.
Identify the asymptotes of $f(x) = \frac{1}{x^2 - 4x + 3}$.
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Vertical asymptotes at $x = 1$ and $x = 3$. Factor $x^2 - 4x + 3 = (x-1)(x-3) = 0$.
Vertical asymptotes at $x = 1$ and $x = 3$. Factor $x^2 - 4x + 3 = (x-1)(x-3) = 0$.
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Identify the vertical asymptote in $f(x) = \frac{1}{x-2}$.
Identify the vertical asymptote in $f(x) = \frac{1}{x-2}$.
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Vertical asymptote at $x = 2$. Denominator equals zero when $x - 2 = 0$.
Vertical asymptote at $x = 2$. Denominator equals zero when $x - 2 = 0$.
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