Limits at Infinity and Horizontal Asymptotes - AP Calculus AB
Card 1 of 30
Determine $\lim_{x \to \infty} \frac{5x - 1}{2x + 3}$.
Determine $\lim_{x \to \infty} \frac{5x - 1}{2x + 3}$.
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$\frac{5}{2}$. Divide by highest power; $\frac{5}{2}$ is the ratio of leading coefficients.
$\frac{5}{2}$. Divide by highest power; $\frac{5}{2}$ is the ratio of leading coefficients.
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What is the horizontal asymptote of $f(x) = \frac{x^2}{x^3 + 1}$?
What is the horizontal asymptote of $f(x) = \frac{x^2}{x^3 + 1}$?
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$y = 0$. Denominator degree exceeds numerator degree, so asymptote is $y = 0$.
$y = 0$. Denominator degree exceeds numerator degree, so asymptote is $y = 0$.
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What is the horizontal asymptote of $f(x) = \frac{4x^3 + 5}{x^3 + 6}$?
What is the horizontal asymptote of $f(x) = \frac{4x^3 + 5}{x^3 + 6}$?
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$y = 4$. Same degree; ratio of leading coefficients is $\frac{4}{1} = 4$.
$y = 4$. Same degree; ratio of leading coefficients is $\frac{4}{1} = 4$.
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Determine $\text{lim}_{x \to \infty} \frac{2x^3 - 5}{3x^3 + 1}$.
Determine $\text{lim}_{x \to \infty} \frac{2x^3 - 5}{3x^3 + 1}$.
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$\frac{2}{3}$. Same degree; ratio of leading coefficients is $\frac{2}{3}$.
$\frac{2}{3}$. Same degree; ratio of leading coefficients is $\frac{2}{3}$.
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Find the horizontal asymptote of $f(x) = \frac{x + 1}{x^2 + 1}$.
Find the horizontal asymptote of $f(x) = \frac{x + 1}{x^2 + 1}$.
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$y = 0$. Denominator degree exceeds numerator degree, so horizontal asymptote is $y = 0$.
$y = 0$. Denominator degree exceeds numerator degree, so horizontal asymptote is $y = 0$.
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State the horizontal asymptote of $f(x) = \frac{8x}{x^2 + 3}$.
State the horizontal asymptote of $f(x) = \frac{8x}{x^2 + 3}$.
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$y = 0$. Denominator degree exceeds numerator degree, so asymptote is $y = 0$.
$y = 0$. Denominator degree exceeds numerator degree, so asymptote is $y = 0$.
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Find the limit as $x$ approaches infinity for $f(x) = \frac{7x^3}{2x^3 + 5}$.
Find the limit as $x$ approaches infinity for $f(x) = \frac{7x^3}{2x^3 + 5}$.
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$\frac{7}{2}$. Same degree; ratio of leading coefficients is $\frac{7}{2}$.
$\frac{7}{2}$. Same degree; ratio of leading coefficients is $\frac{7}{2}$.
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Find the horizontal asymptote of $f(x) = \frac{6x^3}{3x^3 + 1}$.
Find the horizontal asymptote of $f(x) = \frac{6x^3}{3x^3 + 1}$.
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$y = 2$. Same degree; ratio of leading coefficients is $\frac{6}{3} = 2$.
$y = 2$. Same degree; ratio of leading coefficients is $\frac{6}{3} = 2$.
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State the horizontal asymptote of $f(x) = \frac{x^2 - 3}{2x^2 + x + 1}$.
State the horizontal asymptote of $f(x) = \frac{x^2 - 3}{2x^2 + x + 1}$.
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$y = \frac{1}{2}$. Same degree; ratio of leading coefficients is $\frac{1}{2}$.
$y = \frac{1}{2}$. Same degree; ratio of leading coefficients is $\frac{1}{2}$.
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What is the horizontal asymptote of $f(x) = \frac{8x^3 + 1}{4x^3 + 2}$?
What is the horizontal asymptote of $f(x) = \frac{8x^3 + 1}{4x^3 + 2}$?
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$y = 2$. Same degree; ratio of leading coefficients is $\frac{8}{4} = 2$.
$y = 2$. Same degree; ratio of leading coefficients is $\frac{8}{4} = 2$.
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What is the horizontal asymptote of $f(x) = \frac{2x^2 + 3}{x^2 + 5}$?
What is the horizontal asymptote of $f(x) = \frac{2x^2 + 3}{x^2 + 5}$?
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$y = 2$. Same degree numerator and denominator; ratio of leading coefficients is $\frac{2}{1} = 2$.
$y = 2$. Same degree numerator and denominator; ratio of leading coefficients is $\frac{2}{1} = 2$.
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What is the horizontal asymptote of $f(x) = \frac{3x + 2}{2x^2 + 5}$?
What is the horizontal asymptote of $f(x) = \frac{3x + 2}{2x^2 + 5}$?
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$y = 0$. Denominator degree exceeds numerator degree, so asymptote is $y = 0$.
$y = 0$. Denominator degree exceeds numerator degree, so asymptote is $y = 0$.
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What is the limit as $x$ approaches infinity for $f(x) = \frac{3x^3 + 2}{x^2 + 4}$?
What is the limit as $x$ approaches infinity for $f(x) = \frac{3x^3 + 2}{x^2 + 4}$?
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Infinity. Numerator degree exceeds denominator degree, so limit approaches infinity.
Infinity. Numerator degree exceeds denominator degree, so limit approaches infinity.
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Determine $\text{lim}_{x \to -\infty} \frac{-x^3 + 4}{3x^3 - x}$.
Determine $\text{lim}_{x \to -\infty} \frac{-x^3 + 4}{3x^3 - x}$.
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$\frac{-1}{3}$. Same degree; ratio of leading coefficients is $\frac{-1}{3}$.
$\frac{-1}{3}$. Same degree; ratio of leading coefficients is $\frac{-1}{3}$.
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Find the limit as $x$ approaches infinity of $f(x) = \frac{3x^4}{x^4 + 5}$.
Find the limit as $x$ approaches infinity of $f(x) = \frac{3x^4}{x^4 + 5}$.
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$3$. Same degree; ratio of leading coefficients is $\frac{3}{1} = 3$.
$3$. Same degree; ratio of leading coefficients is $\frac{3}{1} = 3$.
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Determine $\text{lim}_{x \to \infty} \frac{8x - 3}{2x + 1}$.
Determine $\text{lim}_{x \to \infty} \frac{8x - 3}{2x + 1}$.
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$4$. Same degree; ratio of leading coefficients is $\frac{8}{2} = 4$.
$4$. Same degree; ratio of leading coefficients is $\frac{8}{2} = 4$.
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State the horizontal asymptote of $f(x) = \frac{7x^2 - x}{3x^2 + 2}$.
State the horizontal asymptote of $f(x) = \frac{7x^2 - x}{3x^2 + 2}$.
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$y = \frac{7}{3}$. Same degree; ratio of leading coefficients is $\frac{7}{3}$.
$y = \frac{7}{3}$. Same degree; ratio of leading coefficients is $\frac{7}{3}$.
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Find the horizontal asymptote of $f(x) = \frac{5x^3 - 2x}{4x^3 + x}$.
Find the horizontal asymptote of $f(x) = \frac{5x^3 - 2x}{4x^3 + x}$.
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$y = \frac{5}{4}$. Same degree; ratio of leading coefficients is $\frac{5}{4}$.
$y = \frac{5}{4}$. Same degree; ratio of leading coefficients is $\frac{5}{4}$.
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Determine $\text{lim}_{x \to -\infty} \frac{x^2 - 2x}{x^2 + x}$.
Determine $\text{lim}_{x \to -\infty} \frac{x^2 - 2x}{x^2 + x}$.
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$1$. Same degree; ratio of leading coefficients is $\frac{1}{1} = 1$.
$1$. Same degree; ratio of leading coefficients is $\frac{1}{1} = 1$.
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What is the horizontal asymptote for $f(x) = \frac{2x^3}{5x^3 - 4}$?
What is the horizontal asymptote for $f(x) = \frac{2x^3}{5x^3 - 4}$?
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$y = \frac{2}{5}$. Same degree; ratio of leading coefficients is $\frac{2}{5}$.
$y = \frac{2}{5}$. Same degree; ratio of leading coefficients is $\frac{2}{5}$.
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Determine $\text{lim}_{x \to \infty} \frac{7x^2 - 5}{2x^2 + 4}$.
Determine $\text{lim}_{x \to \infty} \frac{7x^2 - 5}{2x^2 + 4}$.
Tap to reveal answer
$\frac{7}{2}$. Same degree; ratio of leading coefficients is $\frac{7}{2}$.
$\frac{7}{2}$. Same degree; ratio of leading coefficients is $\frac{7}{2}$.
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State the horizontal asymptote of $f(x) = \frac{x^2 + x}{x^3 + 1}$.
State the horizontal asymptote of $f(x) = \frac{x^2 + x}{x^3 + 1}$.
Tap to reveal answer
$y = 0$. Denominator degree exceeds numerator degree, so asymptote is $y = 0$.
$y = 0$. Denominator degree exceeds numerator degree, so asymptote is $y = 0$.
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What is the limit as $x$ approaches infinity for $f(x) = \frac{6x^2 - 9}{3x^2 + 7}$?
What is the limit as $x$ approaches infinity for $f(x) = \frac{6x^2 - 9}{3x^2 + 7}$?
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$2$. Same degree; ratio of leading coefficients is $\frac{6}{3} = 2$.
$2$. Same degree; ratio of leading coefficients is $\frac{6}{3} = 2$.
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What is the horizontal asymptote of $f(x) = \frac{x^3 - 1}{x^2 + 3x}$?
What is the horizontal asymptote of $f(x) = \frac{x^3 - 1}{x^2 + 3x}$?
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None. Numerator degree exceeds denominator degree, so no horizontal asymptote exists.
None. Numerator degree exceeds denominator degree, so no horizontal asymptote exists.
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Find the limit as $x$ approaches infinity of $f(x) = \frac{4x^2}{2x^2 + 3}$.
Find the limit as $x$ approaches infinity of $f(x) = \frac{4x^2}{2x^2 + 3}$.
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$2$. Same degree; ratio of leading coefficients is $\frac{4}{2} = 2$.
$2$. Same degree; ratio of leading coefficients is $\frac{4}{2} = 2$.
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State the horizontal asymptote of $f(x) = \frac{5x^2}{3x^2 + 2}$.
State the horizontal asymptote of $f(x) = \frac{5x^2}{3x^2 + 2}$.
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$y = \frac{5}{3}$. Same degree; ratio of leading coefficients is $\frac{5}{3}$.
$y = \frac{5}{3}$. Same degree; ratio of leading coefficients is $\frac{5}{3}$.
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What is the horizontal asymptote of $f(x) = \frac{7x}{2x + 3}$?
What is the horizontal asymptote of $f(x) = \frac{7x}{2x + 3}$?
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$y = \frac{7}{2}$. Same degree polynomials; ratio of leading coefficients is $\frac{7}{2}$.
$y = \frac{7}{2}$. Same degree polynomials; ratio of leading coefficients is $\frac{7}{2}$.
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Determine $\text{lim}_{x \to \infty} \frac{5x - 1}{2x + 3}$.
Determine $\text{lim}_{x \to \infty} \frac{5x - 1}{2x + 3}$.
Tap to reveal answer
$\frac{5}{2}$. Divide by highest power; $\frac{5}{2}$ is the ratio of leading coefficients.
$\frac{5}{2}$. Divide by highest power; $\frac{5}{2}$ is the ratio of leading coefficients.
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What is the horizontal asymptote of $f(x) = \frac{4x^3 + 5}{x^3 + 6}$?
What is the horizontal asymptote of $f(x) = \frac{4x^3 + 5}{x^3 + 6}$?
Tap to reveal answer
$y = 4$. Same degree; ratio of leading coefficients is $\frac{4}{1} = 4$.
$y = 4$. Same degree; ratio of leading coefficients is $\frac{4}{1} = 4$.
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What is the limit as $x$ approaches infinity for $f(x) = \frac{6x^2 - 9}{3x^2 + 7}$?
What is the limit as $x$ approaches infinity for $f(x) = \frac{6x^2 - 9}{3x^2 + 7}$?
Tap to reveal answer
$2$. Same degree; ratio of leading coefficients is $\frac{6}{3} = 2$.
$2$. Same degree; ratio of leading coefficients is $\frac{6}{3} = 2$.
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