Limits at Infinity and Horizontal Asymptotes - AP Calculus BC
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What is the horizontal asymptote of $f(x) = \frac{4x^4}{x^3 + 2}$?
What is the horizontal asymptote of $f(x) = \frac{4x^4}{x^3 + 2}$?
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None. Numerator degree exceeds denominator degree.
None. Numerator degree exceeds denominator degree.
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State the horizontal asymptote for $f(x) = \frac{x^3 - x}{3x^3 + 5}$.
State the horizontal asymptote for $f(x) = \frac{x^3 - x}{3x^3 + 5}$.
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$y = \frac{1}{3}$. Cubic leading coefficients: $\frac{1}{3}$.
$y = \frac{1}{3}$. Cubic leading coefficients: $\frac{1}{3}$.
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Determine the horizontal asymptote of $f(x) = \frac{3x^2 + 5}{4x^2 + 7}$.
Determine the horizontal asymptote of $f(x) = \frac{3x^2 + 5}{4x^2 + 7}$.
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$y = \frac{3}{4}$. Quadratic coefficients: $\frac{3}{4}$.
$y = \frac{3}{4}$. Quadratic coefficients: $\frac{3}{4}$.
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Determine the horizontal asymptote of $f(x) = \frac{4x}{x^3 + 1}$.
Determine the horizontal asymptote of $f(x) = \frac{4x}{x^3 + 1}$.
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$y = 0$. Numerator degree less than denominator degree.
$y = 0$. Numerator degree less than denominator degree.
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Find the horizontal asymptote for $f(x) = \frac{2x^4 + 3x}{x^4 + x^2}$.
Find the horizontal asymptote for $f(x) = \frac{2x^4 + 3x}{x^4 + x^2}$.
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$y = 2$. Fourth-degree terms dominate: $\frac{2}{1} = 2$.
$y = 2$. Fourth-degree terms dominate: $\frac{2}{1} = 2$.
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Identify the horizontal asymptote of $f(x) = \frac{7x^3 + 5}{3x^3 - 2}$.
Identify the horizontal asymptote of $f(x) = \frac{7x^3 + 5}{3x^3 - 2}$.
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$y = \frac{7}{3}$. Equal degree cubic polynomials: $\frac{7}{3}$.
$y = \frac{7}{3}$. Equal degree cubic polynomials: $\frac{7}{3}$.
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What is the horizontal asymptote of $f(x) = \frac{5x^3}{2x^3 + 7}$?
What is the horizontal asymptote of $f(x) = \frac{5x^3}{2x^3 + 7}$?
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$y = \frac{5}{2}$. Ratio of leading coefficients for equal-degree polynomials.
$y = \frac{5}{2}$. Ratio of leading coefficients for equal-degree polynomials.
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What is the horizontal asymptote of $f(x) = \frac{2x^2 - 5x}{5x^2 + x}$?
What is the horizontal asymptote of $f(x) = \frac{2x^2 - 5x}{5x^2 + x}$?
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$y = \frac{2}{5}$. Quadratic terms: $\frac{2}{5}$.
$y = \frac{2}{5}$. Quadratic terms: $\frac{2}{5}$.
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State the horizontal asymptote of $f(x) = \frac{x^2 + 2}{2x^2 - 3}$.
State the horizontal asymptote of $f(x) = \frac{x^2 + 2}{2x^2 - 3}$.
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$y = \frac{1}{2}$. Leading coefficient ratio: $\frac{1}{2}$.
$y = \frac{1}{2}$. Leading coefficient ratio: $\frac{1}{2}$.
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What is the horizontal asymptote of $f(x) = \frac{2x^3}{x^2 + x}$?
What is the horizontal asymptote of $f(x) = \frac{2x^3}{x^2 + x}$?
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None. Numerator degree exceeds denominator degree.
None. Numerator degree exceeds denominator degree.
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What is the horizontal asymptote of $f(x) = \frac{x^3 + 2}{4x^3 + x}$?
What is the horizontal asymptote of $f(x) = \frac{x^3 + 2}{4x^3 + x}$?
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$y = \frac{1}{4}$. Cubic coefficients: $\frac{1}{4}$.
$y = \frac{1}{4}$. Cubic coefficients: $\frac{1}{4}$.
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State the horizontal asymptote for $f(x) = \frac{2x}{x^3 + 5}$.
State the horizontal asymptote for $f(x) = \frac{2x}{x^3 + 5}$.
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$y = 0$. Linear over cubic approaches 0.
$y = 0$. Linear over cubic approaches 0.
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Find the horizontal asymptote for $f(x) = \frac{3x^4 - 2x}{2x^4 + 5}$.
Find the horizontal asymptote for $f(x) = \frac{3x^4 - 2x}{2x^4 + 5}$.
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$y = \frac{3}{2}$. Fourth-degree leading coefficients: $\frac{3}{2}$.
$y = \frac{3}{2}$. Fourth-degree leading coefficients: $\frac{3}{2}$.
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What is the horizontal asymptote of $f(x) = \frac{5x}{x^2 + x}$?
What is the horizontal asymptote of $f(x) = \frac{5x}{x^2 + x}$?
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$y = 0$. Linear over quadratic approaches 0.
$y = 0$. Linear over quadratic approaches 0.
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Find the horizontal asymptote for $f(x) = \frac{4x^2 - x}{x^2 + x}$.
Find the horizontal asymptote for $f(x) = \frac{4x^2 - x}{x^2 + x}$.
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$y = 4$. Quadratic coefficients: $\frac{4}{1} = 4$.
$y = 4$. Quadratic coefficients: $\frac{4}{1} = 4$.
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What is the horizontal asymptote of $f(x) = \frac{x^2 + 7x}{3x^2 - 9}$?
What is the horizontal asymptote of $f(x) = \frac{x^2 + 7x}{3x^2 - 9}$?
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$y = \frac{1}{3}$. Quadratic leading coefficients: $\frac{1}{3}$.
$y = \frac{1}{3}$. Quadratic leading coefficients: $\frac{1}{3}$.
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What is the horizontal asymptote of $f(x) = \frac{2x^2 + 4x}{x^2}$?
What is the horizontal asymptote of $f(x) = \frac{2x^2 + 4x}{x^2}$?
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$y = 2$. Simplify to $\frac{2x^2 + 4x}{x^2} = 2 + \frac{4}{x}$.
$y = 2$. Simplify to $\frac{2x^2 + 4x}{x^2} = 2 + \frac{4}{x}$.
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Determine the horizontal asymptote of $f(x) = \frac{7x^3}{3x^3 + 2x}$.
Determine the horizontal asymptote of $f(x) = \frac{7x^3}{3x^3 + 2x}$.
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$y = \frac{7}{3}$. Cubic terms: $\frac{7}{3}$.
$y = \frac{7}{3}$. Cubic terms: $\frac{7}{3}$.
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Find the horizontal asymptote for $f(x) = \frac{x^3}{x^3 + x}$.
Find the horizontal asymptote for $f(x) = \frac{x^3}{x^3 + x}$.
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$y = 1$. Cubic terms: $\frac{1}{1} = 1$.
$y = 1$. Cubic terms: $\frac{1}{1} = 1$.
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What is the limit of $f(x) = \frac{4x^2 - x + 6}{x^2 + x - 12}$ as $x$ approaches infinity?
What is the limit of $f(x) = \frac{4x^2 - x + 6}{x^2 + x - 12}$ as $x$ approaches infinity?
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$4$. Leading coefficient ratio when degrees match.
$4$. Leading coefficient ratio when degrees match.
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Identify the horizontal asymptote of $f(x) = \frac{x}{x^3 + 1}$.
Identify the horizontal asymptote of $f(x) = \frac{x}{x^3 + 1}$.
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$y = 0$. Linear over cubic approaches 0.
$y = 0$. Linear over cubic approaches 0.
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What is the horizontal asymptote of $f(x) = \frac{4x}{x^2 + 1}$?
What is the horizontal asymptote of $f(x) = \frac{4x}{x^2 + 1}$?
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$y = 0$. Linear over quadratic approaches 0.
$y = 0$. Linear over quadratic approaches 0.
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Identify the horizontal asymptote of $f(x) = \frac{x^2 + 3}{2x^2 - 1}$.
Identify the horizontal asymptote of $f(x) = \frac{x^2 + 3}{2x^2 - 1}$.
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$y = \frac{1}{2}$. Quadratic coefficients: $\frac{1}{2}$.
$y = \frac{1}{2}$. Quadratic coefficients: $\frac{1}{2}$.
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What is the horizontal asymptote of $f(x) = \frac{x^3 + 5}{4x^4 + 1}$?
What is the horizontal asymptote of $f(x) = \frac{x^3 + 5}{4x^4 + 1}$?
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$y = 0$. Numerator degree less than denominator degree.
$y = 0$. Numerator degree less than denominator degree.
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Determine the horizontal asymptote of $f(x) = \frac{3x^2 + 2}{x^2 - x}$.
Determine the horizontal asymptote of $f(x) = \frac{3x^2 + 2}{x^2 - x}$.
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$y = 3$. Quadratic terms: $\frac{3}{1} = 3$.
$y = 3$. Quadratic terms: $\frac{3}{1} = 3$.
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What is the horizontal asymptote of $f(x) = \frac{5x^3}{2x^2 + x}$?
What is the horizontal asymptote of $f(x) = \frac{5x^3}{2x^2 + x}$?
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None. Numerator degree exceeds denominator degree.
None. Numerator degree exceeds denominator degree.
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State the horizontal asymptote for $f(x) = \frac{x^2 - 4}{x^2 + 5}$.
State the horizontal asymptote for $f(x) = \frac{x^2 - 4}{x^2 + 5}$.
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$y = 1$. Equal-degree quadratics: $\frac{1}{1} = 1$.
$y = 1$. Equal-degree quadratics: $\frac{1}{1} = 1$.
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What is the limit of $f(x) = \frac{6x^3}{3x^4 + 2}$ as $x$ approaches infinity?
What is the limit of $f(x) = \frac{6x^3}{3x^4 + 2}$ as $x$ approaches infinity?
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$0$. Numerator degree less than denominator degree.
$0$. Numerator degree less than denominator degree.
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What is the horizontal asymptote of $f(x) = \frac{2x}{x^2 + 1}$?
What is the horizontal asymptote of $f(x) = \frac{2x}{x^2 + 1}$?
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$y = 0$. Linear over quadratic approaches 0.
$y = 0$. Linear over quadratic approaches 0.
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Find the horizontal asymptote for $f(x) = \frac{4x^3 + 1}{x^3 - 2}$.
Find the horizontal asymptote for $f(x) = \frac{4x^3 + 1}{x^3 - 2}$.
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$y = 4$. Cubic terms give $\frac{4}{1} = 4$.
$y = 4$. Cubic terms give $\frac{4}{1} = 4$.
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