AP Calculus AB Flashcards: Limits At Infinity And Horizontal Asymptotes

Study Limits At Infinity And Horizontal 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

Limits At Infinity And Horizontal Asymptotes

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QUESTION

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What is the limit as $x$ approaches infinity for $f(x) = \frac{2x + 1}{3x + 4}$ ?

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ANSWER

$\frac{2}{3}$ . Same degree; ratio of leading coefficients is $\frac{2}{3}$ .

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What this deck covers

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

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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: What is the limit as $x$ approaches infinity for $f(x) = \frac{2x + 1}{3x + 4}$ ?

Answer: $\frac{2}{3}$ . Same degree; ratio of leading coefficients is $\frac{2}{3}$ .

Flashcard 2: Find the horizontal asymptote of $f(x) = \frac{3x + 2}{4x + 5}$ .

Answer: $y = \frac{3}{4}$ . Same degree; ratio of leading coefficients is $\frac{3}{4}$ .

Flashcard 3: Find the limit as $x$ approaches infinity for $f(x) = \frac{7x^3}{2x^3 + 5}$ .

Answer: $\frac{7}{2}$ . Same degree; ratio of leading coefficients is $\frac{7}{2}$ .

Flashcard 4: State the horizontal asymptote of $f(x) = \frac{5x}{x^3 + 2}$ .

Answer: $y = 0$ . Denominator degree exceeds numerator degree, so asymptote is $y = 0$ .

Flashcard 5: Determine $\text{lim}_{x \to \infty} \frac{5x^2 + 3}{3x^2 - 2}$ .

Answer: $\frac{5}{3}$ . Same degree; ratio of leading coefficients is $\frac{5}{3}$ .

Flashcard 6: Determine $\text{lim}_{x \to \infty} \frac{8x - 3}{2x + 1}$ .

Answer: $4$ . Same degree; ratio of leading coefficients is $\frac{8}{2} = 4$ .

Flashcard 7: What is the horizontal asymptote of $f(x) = \frac{2x^3 + 1}{5x^3 + 2}$ ?

Answer: $y = \frac{2}{5}$ . Same degree; ratio of leading coefficients is $\frac{2}{5}$ .

Flashcard 8: What is the horizontal asymptote of $f(x) = \frac{3x + 2}{2x^2 + 5}$ ?

Answer: $y = 0$ . Denominator degree exceeds numerator degree, so asymptote is $y = 0$ .

Flashcard 9: What is the limit as $x$ approaches $\infty$ for $f(x) = \frac{3x^3 + 2}{x^2 + 4}$ ?

Answer: $\infty$ . Numerator degree exceeds denominator degree, so limit approaches $\infty$ .

Flashcard 10: What is the horizontal asymptote of $f(x) = \frac{x^3 - 1}{x^2 + 3x}$ ?

Answer: None. Numerator degree exceeds denominator degree, so no horizontal asymptote exists.

Flashcard 11: What is the horizontal asymptote of $f(x) = \frac{x^2}{x^3 + 1}$ ?

Answer: $y = 0$ . Denominator degree exceeds numerator degree, so asymptote is $y = 0$ .

Flashcard 12: Determine $\text{lim}_{x \to -\infty} \frac{-x^3 + 4}{3x^3 - x}$ .

Answer: $\frac{-1}{3}$ . Same degree; ratio of leading coefficients is $\frac{-1}{3}$ .

Flashcard 13: Determine $\text{lim}_{x \to \infty} \frac{8x - 3}{2x + 1}$ .

Answer: $4$ . Same degree; ratio of leading coefficients is $\frac{8}{2} = 4$ .

Flashcard 14: What is the horizontal asymptote of $f(x) = \frac{3x + 2}{2x^2 + 5}$ ?

Answer: $y = 0$ . Denominator degree exceeds numerator degree, so asymptote is $y = 0$ .

Flashcard 15: State the horizontal asymptote of $f(x) = \frac{x^2 - 3}{2x^2 + x + 1}$ .

Answer: $y = \frac{1}{2}$ . Same degree; ratio of leading coefficients is $\frac{1}{2}$ .

Flashcard 16: What is the horizontal asymptote of $f(x) = \frac{7x}{2x + 3}$ ?

Answer: $y = \frac{7}{2}$ . Same degree polynomials; ratio of leading coefficients is $\frac{7}{2}$ .

Flashcard 17: State the horizontal asymptote of $f(x) = \frac{7x^2 - x}{3x^2 + 2}$ .

Answer: $y = \frac{7}{3}$ . Same degree; ratio of leading coefficients is $\frac{7}{3}$ .

Flashcard 18: What is the horizontal asymptote for $f(x) = \frac{2x^3}{5x^3 - 4}$ ?

Answer: $y = \frac{2}{5}$ . Same degree; ratio of leading coefficients is $\frac{2}{5}$ .

Flashcard 19: Find the limit as $x$ approaches infinity for $f(x) = \frac{7x^3}{2x^3 + 5}$ .

Answer: $\frac{7}{2}$ . Same degree; ratio of leading coefficients is $\frac{7}{2}$ .

Flashcard 20: Find the limit as $x$ approaches infinity for $f(x) = \frac{x^2 - 1}{2x^2 + x}$ .

Answer: $\frac{1}{2}$ . Same degree; ratio of leading coefficients is $\frac{1}{2}$ .

Flashcard 21: Determine $\text{lim}_{x \to \infty} \frac{2x^3 - 5}{3x^3 + 1}$ .

Answer: $\frac{2}{3}$ . Same degree; ratio of leading coefficients is $\frac{2}{3}$ .

Flashcard 22: Determine $\text{lim}_{x \to \infty} \frac{7x^2 - 5}{2x^2 + 4}$ .

Answer: $\frac{7}{2}$ . Same degree; ratio of leading coefficients is $\frac{7}{2}$ .

Flashcard 23: What is the horizontal asymptote of $f(x) = \frac{x^2}{x^3 + 1}$ ?

Answer: $y = 0$ . Denominator degree exceeds numerator degree, so asymptote is $y = 0$ .

Flashcard 24: What is the horizontal asymptote for $f(x) = \frac{4x}{x + 5}$ ?

Answer: $y = 4$ . Same degree; ratio of leading coefficients is $\frac{4}{1} = 4$ .

Flashcard 25: State the horizontal asymptote of $f(x) = \frac{8x}{x^2 + 3}$ .

Answer: $y = 0$ . Denominator degree exceeds numerator degree, so asymptote is $y = 0$ .

Flashcard 26: Find the limit as $x$ approaches infinity for $f(x) = \frac{x^2 - 1}{2x^2 + x}$ .

Answer: $\frac{1}{2}$ . Same degree; ratio of leading coefficients is $\frac{1}{2}$ .

Flashcard 27: Find the horizontal asymptote of $f(x) = \frac{x + 1}{x^2 + 1}$ .

Answer: $y = 0$ . Denominator degree exceeds numerator degree, so horizontal asymptote is $y = 0$ .

Flashcard 28: What is the horizontal asymptote of $f(x) = \frac{2x^2 + 3}{x^2 + 5}$ ?

Answer: $y = 2$ . Same degree numerator and denominator; ratio of leading coefficients is $\frac{2}{1} = 2$ .

Flashcard 29: State the horizontal asymptote of $f(x) = \frac{x^2 - 3}{2x^2 + x + 1}$ .

Answer: $y = \frac{1}{2}$ . Same degree; ratio of leading coefficients is $\frac{1}{2}$ .

Flashcard 30: What is the horizontal asymptote for $f(x) = \frac{2x^2 + 3x}{5x^2 + 1}$ ?

Answer: $y = \frac{2}{5}$ . Same degree; ratio of leading coefficients is $\frac{2}{5}$ .

Flashcard 31: Determine $\text{lim}_{x \to -\infty} \frac{x^2 - 2x}{x^2 + x}$ .

Answer: $1$ . Same degree; ratio of leading coefficients is $\frac{1}{1} = 1$ .

Flashcard 32: What is the horizontal asymptote of $f(x) = \frac{8x^3 + 1}{4x^3 + 2}$ ?

Answer: $y = 2$ . Same degree; ratio of leading coefficients is $\frac{8}{4} = 2$ .

Flashcard 33: What is the horizontal asymptote of $f(x) = \frac{x + 4}{x^2 + 2x}$ ?

Answer: $y = 0$ . Denominator degree exceeds numerator degree, so asymptote is $y = 0$ .

Flashcard 34: Find the limit as $x$ approaches infinity of $f(x) = \frac{3x^4}{x^4 + 5}$ .

Answer: $3$ . Same degree; ratio of leading coefficients is $\frac{3}{1} = 3$ .

Flashcard 35: What is the horizontal asymptote of $f(x) = \frac{8x^3 + 1}{4x^3 + 2}$ ?

Answer: $y = 2$ . Same degree; ratio of leading coefficients is $\frac{8}{4} = 2$ .

Flashcard 36: Determine $\text{lim}_{x \to -\infty} \frac{6x - 1}{2x + 3}$ .

Answer: $3$ . Same degree; ratio of leading coefficients is $\frac{6}{2} = 3$ .

Flashcard 37: Determine $\text{lim}_{x \to \infty} \frac{2x^3 - 5}{3x^3 + 1}$ .

Answer: $\frac{2}{3}$ . Same degree; ratio of leading coefficients is $\frac{2}{3}$ .

Flashcard 38: What is the horizontal asymptote of $f(x) = \frac{4x^3 + 5}{x^3 + 6}$ ?

Answer: $y = 4$ . Same degree; ratio of leading coefficients is $\frac{4}{1} = 4$ .

Flashcard 39: What is the limit as $x$ approaches infinity for $f(x) = \frac{6x^2 - 9}{3x^2 + 7}$ ?

Answer: $2$ . Same degree; ratio of leading coefficients is $\frac{6}{3} = 2$ .

Flashcard 40: Find the horizontal asymptote of $f(x) = \frac{3x + 2}{4x + 5}$ .

Answer: $y = \frac{3}{4}$ . Same degree; ratio of leading coefficients is $\frac{3}{4}$ .

Flashcard 41: What is the horizontal asymptote of $f(x) = \frac{2x^3 + 1}{5x^3 + 2}$ ?

Answer: $y = \frac{2}{5}$ . Same degree; ratio of leading coefficients is $\frac{2}{5}$ .

Flashcard 42: Find the limit as $x$ approaches infinity of $f(x) = \frac{4x^2}{2x^2 + 3}$ .

Answer: $2$ . Same degree; ratio of leading coefficients is $\frac{4}{2} = 2$ .

Flashcard 43: State the horizontal asymptote of $f(x) = \frac{7x^2 - x}{3x^2 + 2}$ .

Answer: $y = \frac{7}{3}$ . Same degree; ratio of leading coefficients is $\frac{7}{3}$ .

Flashcard 44: What is the horizontal asymptote for $f(x) = \frac{9x + 7}{3x + 1}$ ?

Answer: $y = 3$ . Same degree; ratio of leading coefficients is $\frac{9}{3} = 3$ .

Flashcard 45: What is the limit as $x$ approaches infinity for $f(x) = \frac{3x^3 + 2}{x^2 + 4}$ ?

Answer: Infinity. Numerator degree exceeds denominator degree, so limit approaches infinity.

Flashcard 46: What is the limit as $x$ approaches infinity for $f(x) = \frac{6x^2 - 9}{3x^2 + 7}$ ?

Answer: $2$ . Same degree; ratio of leading coefficients is $\frac{6}{3} = 2$ .

Flashcard 47: State the horizontal asymptote of $f(x) = \frac{5x^2}{3x^2 + 2}$ .

Answer: $y = \frac{5}{3}$ . Same degree; ratio of leading coefficients is $\frac{5}{3}$ .

Flashcard 48: Find the limit as $x$ approaches infinity of $f(x) = \frac{3x^4}{x^4 + 5}$ .

Answer: $3$ . Same degree; ratio of leading coefficients is $\frac{3}{1} = 3$ .

Flashcard 49: Determine $\text{lim}_{x \to \infty} \frac{5x^2 + 3}{3x^2 - 2}$ .

Answer: $\frac{5}{3}$ . Same degree; ratio of leading coefficients is $\frac{5}{3}$ .

Flashcard 50: State the horizontal asymptote of $f(x) = \frac{x^2 + x}{x^3 + 1}$ .

Answer: $y = 0$ . Denominator degree exceeds numerator degree, so asymptote is $y = 0$ .

Flashcard 51: What is the horizontal asymptote for $f(x) = \frac{4x^2 - 3}{x^2 + 5x}$ ?

Answer: $y = 4$ . Same degree; ratio of leading coefficients is $\frac{4}{1} = 4$ .

Flashcard 52: Find the horizontal asymptote of $f(x) = \frac{6x^3}{3x^3 + 1}$ .

Answer: $y = 2$ . Same degree; ratio of leading coefficients is $\frac{6}{3} = 2$ .

Flashcard 53: What is the horizontal asymptote of $f(x) = \frac{2x^2 + 3}{x^2 + 5}$ ?

Answer: $y = 2$ . Same degree numerator and denominator; ratio of leading coefficients is $\frac{2}{1} = 2$ .

Flashcard 54: Determine $\text{lim}_{x \to -\infty} \frac{6x - 1}{2x + 3}$ .

Answer: $3$ . Same degree; ratio of leading coefficients is $\frac{6}{2} = 3$ .

Flashcard 55: What is the horizontal asymptote for $f(x) = \frac{9x + 7}{3x + 1}$ ?

Answer: $y = 3$ . Same degree; ratio of leading coefficients is $\frac{9}{3} = 3$ .

Flashcard 56: State the horizontal asymptote of $f(x) = \frac{x^2 + x}{x^3 + 1}$ .

Answer: $y = 0$ . Denominator degree exceeds numerator degree, so asymptote is $y = 0$ .

Flashcard 57: Find the horizontal asymptote of $f(x) = \frac{6x^3}{3x^3 + 1}$ .

Answer: $y = 2$ . Same degree; ratio of leading coefficients is $\frac{6}{3} = 2$ .

Flashcard 58: Determine $\text{lim}_{x \to -\infty} \frac{x^2 - 2x}{x^2 + x}$ .

Answer: $1$ . Same degree; ratio of leading coefficients is $\frac{1}{1} = 1$ .

Flashcard 59: What is the horizontal asymptote of $f(x) = \frac{7x}{2x + 3}$ ?

Answer: $y = \frac{7}{2}$ . Same degree polynomials; ratio of leading coefficients is $\frac{7}{2}$ .

Flashcard 60: What is the horizontal asymptote for $f(x) = \frac{2x^2 + 3x}{5x^2 + 1}$ ?

Answer: $y = \frac{2}{5}$ . Same degree; ratio of leading coefficients is $\frac{2}{5}$ .

Flashcard 61: What is the horizontal asymptote for $f(x) = \frac{2x^3}{5x^3 - 4}$ ?

Answer: $y = \frac{2}{5}$ . Same degree; ratio of leading coefficients is $\frac{2}{5}$ .

Flashcard 62: Find the limit as $x$ approaches infinity of $f(x) = \frac{4x^2}{2x^2 + 3}$ .

Answer: $2$ . Same degree; ratio of leading coefficients is $\frac{4}{2} = 2$ .

Flashcard 63: Determine $\text{lim}_{x \to \infty} \frac{5x - 1}{2x + 3}$ .

Answer: $\frac{5}{2}$ . Divide by highest power; $\frac{5}{2}$ is the ratio of leading coefficients.

Flashcard 64: What is the limit as $x$ approaches infinity for $f(x) = \frac{2x + 1}{3x + 4}$ ?

Answer: $\frac{2}{3}$ . Same degree; ratio of leading coefficients is $\frac{2}{3}$ .

Flashcard 65: Determine $\text{lim}_{x \to \infty} \frac{7x^2 - 5}{2x^2 + 4}$ .

Answer: $\frac{7}{2}$ . Same degree; ratio of leading coefficients is $\frac{7}{2}$ .

Flashcard 66: Determine $\lim_{x \to \infty} \frac{5x - 1}{2x + 3}$ .

Answer: $\frac{5}{2}$ . Divide by highest power; $\frac{5}{2}$ is the ratio of leading coefficients.

Flashcard 67: Find the horizontal asymptote of $f(x) = \frac{x + 1}{x^2 + 1}$ .

Answer: $y = 0$ . Denominator degree exceeds numerator degree, so horizontal asymptote is $y = 0$ .

Flashcard 68: What is the horizontal asymptote for $f(x) = \frac{4x}{x + 5}$ ?

Answer: $y = 4$ . Same degree; ratio of leading coefficients is $\frac{4}{1} = 4$ .

Flashcard 69: What is the horizontal asymptote of $f(x) = \frac{x^3 - 1}{x^2 + 3x}$ ?

Answer: None. Numerator degree exceeds denominator degree, so no horizontal asymptote exists.

Flashcard 70: State the horizontal asymptote of $f(x) = \frac{5x}{x^3 + 2}$ .

Answer: $y = 0$ . Denominator degree exceeds numerator degree, so asymptote is $y = 0$ .

Flashcard 71: State the horizontal asymptote of $f(x) = \frac{5x^2}{3x^2 + 2}$ .

Answer: $y = \frac{5}{3}$ . Same degree; ratio of leading coefficients is $\frac{5}{3}$ .

Flashcard 72: Determine $\text{lim}_{x \to -\infty} \frac{-x^3 + 4}{3x^3 - x}$ .

Answer: $\frac{-1}{3}$ . Same degree; ratio of leading coefficients is $\frac{-1}{3}$ .

Flashcard 73: What is the horizontal asymptote of $f(x) = \frac{4x^3 + 5}{x^3 + 6}$ ?

Answer: $y = 4$ . Same degree; ratio of leading coefficients is $\frac{4}{1} = 4$ .

Flashcard 74: What is the horizontal asymptote for $f(x) = \frac{4x^2 - 3}{x^2 + 5x}$ ?

Answer: $y = 4$ . Same degree; ratio of leading coefficients is $\frac{4}{1} = 4$ .

Flashcard 75: State the horizontal asymptote of $f(x) = \frac{8x}{x^2 + 3}$ .

Answer: $y = 0$ . Denominator degree exceeds numerator degree, so asymptote is $y = 0$ .

Flashcard 76: What is the horizontal asymptote of $f(x) = \frac{x + 4}{x^2 + 2x}$ ?

Answer: $y = 0$ . Denominator degree exceeds numerator degree, so asymptote is $y = 0$ .

Flashcard 77: Find the horizontal asymptote of $f(x) = \frac{5x^3 - 2x}{4x^3 + x}$ .

Answer: $y = \frac{5}{4}$ . Same degree; ratio of leading coefficients is $\frac{5}{4}$ .

Flashcard 78: Find the horizontal asymptote of $f(x) = \frac{5x^3 - 2x}{4x^3 + x}$ .

Answer: $y = \frac{5}{4}$ . Same degree; ratio of leading coefficients is $\frac{5}{4}$ .

AP Calculus AB Flashcards: Limits At Infinity And Horizontal Asymptotes

Limits At Infinity And Horizontal Asymptotes

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Flashcard 1: What is the limit as xxx approaches infinity for f(x)=2x+13x+4f(x) = \frac{2x + 1}{3x + 4}f(x)=3x+42x+1​?

Flashcard 2: Find the horizontal asymptote of f(x)=3x+24x+5f(x) = \frac{3x + 2}{4x + 5}f(x)=4x+53x+2​.

Flashcard 3: Find the limit as xxx approaches infinity for f(x)=7x32x3+5f(x) = \frac{7x^3}{2x^3 + 5}f(x)=2x3+57x3​.

Flashcard 4: State the horizontal asymptote of f(x)=5xx3+2f(x) = \frac{5x}{x^3 + 2}f(x)=x3+25x​.

Flashcard 5: Determine limx→∞5x2+33x2−2\text{lim}_{x \to \infty} \frac{5x^2 + 3}{3x^2 - 2}limx→∞​3x2−25x2+3​.

Flashcard 6: Determine limx→∞8x−32x+1\text{lim}_{x \to \infty} \frac{8x - 3}{2x + 1}limx→∞​2x+18x−3​.

Flashcard 7: What is the horizontal asymptote of f(x)=2x3+15x3+2f(x) = \frac{2x^3 + 1}{5x^3 + 2}f(x)=5x3+22x3+1​?

Flashcard 8: What is the horizontal asymptote of f(x)=3x+22x2+5f(x) = \frac{3x + 2}{2x^2 + 5}f(x)=2x2+53x+2​?

Flashcard 9: What is the limit as xxx approaches ∞\infty∞ for f(x)=3x3+2x2+4f(x) = \frac{3x^3 + 2}{x^2 + 4}f(x)=x2+43x3+2​?

Flashcard 10: What is the horizontal asymptote of f(x)=x3−1x2+3xf(x) = \frac{x^3 - 1}{x^2 + 3x}f(x)=x2+3xx3−1​?

Flashcard 11: What is the horizontal asymptote of f(x)=x2x3+1f(x) = \frac{x^2}{x^3 + 1}f(x)=x3+1x2​?

Flashcard 12: Determine limx→−∞−x3+43x3−x\text{lim}_{x \to -\infty} \frac{-x^3 + 4}{3x^3 - x}limx→−∞​3x3−x−x3+4​.

Flashcard 13: Determine limx→∞8x−32x+1\text{lim}_{x \to \infty} \frac{8x - 3}{2x + 1}limx→∞​2x+18x−3​.

Flashcard 14: What is the horizontal asymptote of f(x)=3x+22x2+5f(x) = \frac{3x + 2}{2x^2 + 5}f(x)=2x2+53x+2​?

Flashcard 15: State the horizontal asymptote of f(x)=x2−32x2+x+1f(x) = \frac{x^2 - 3}{2x^2 + x + 1}f(x)=2x2+x+1x2−3​.

Flashcard 16: What is the horizontal asymptote of f(x)=7x2x+3f(x) = \frac{7x}{2x + 3}f(x)=2x+37x​?

Flashcard 17: State the horizontal asymptote of f(x)=7x2−x3x2+2f(x) = \frac{7x^2 - x}{3x^2 + 2}f(x)=3x2+27x2−x​.

Flashcard 18: What is the horizontal asymptote for f(x)=2x35x3−4f(x) = \frac{2x^3}{5x^3 - 4}f(x)=5x3−42x3​?

Flashcard 19: Find the limit as xxx approaches infinity for f(x)=7x32x3+5f(x) = \frac{7x^3}{2x^3 + 5}f(x)=2x3+57x3​.

Flashcard 20: Find the limit as xxx approaches infinity for f(x)=x2−12x2+xf(x) = \frac{x^2 - 1}{2x^2 + x}f(x)=2x2+xx2−1​.

Flashcard 21: Determine limx→∞2x3−53x3+1\text{lim}_{x \to \infty} \frac{2x^3 - 5}{3x^3 + 1}limx→∞​3x3+12x3−5​.

Flashcard 22: Determine limx→∞7x2−52x2+4\text{lim}_{x \to \infty} \frac{7x^2 - 5}{2x^2 + 4}limx→∞​2x2+47x2−5​.

Flashcard 23: What is the horizontal asymptote of f(x)=x2x3+1f(x) = \frac{x^2}{x^3 + 1}f(x)=x3+1x2​?

Flashcard 24: What is the horizontal asymptote for f(x)=4xx+5f(x) = \frac{4x}{x + 5}f(x)=x+54x​?

Flashcard 25: State the horizontal asymptote of f(x)=8xx2+3f(x) = \frac{8x}{x^2 + 3}f(x)=x2+38x​.

Flashcard 26: Find the limit as xxx approaches infinity for f(x)=x2−12x2+xf(x) = \frac{x^2 - 1}{2x^2 + x}f(x)=2x2+xx2−1​.

Flashcard 27: Find the horizontal asymptote of f(x)=x+1x2+1f(x) = \frac{x + 1}{x^2 + 1}f(x)=x2+1x+1​.

Flashcard 28: What is the horizontal asymptote of f(x)=2x2+3x2+5f(x) = \frac{2x^2 + 3}{x^2 + 5}f(x)=x2+52x2+3​?

Flashcard 29: State the horizontal asymptote of f(x)=x2−32x2+x+1f(x) = \frac{x^2 - 3}{2x^2 + x + 1}f(x)=2x2+x+1x2−3​.

Flashcard 30: What is the horizontal asymptote for f(x)=2x2+3x5x2+1f(x) = \frac{2x^2 + 3x}{5x^2 + 1}f(x)=5x2+12x2+3x​?

Flashcard 31: Determine limx→−∞x2−2xx2+x\text{lim}_{x \to -\infty} \frac{x^2 - 2x}{x^2 + x}limx→−∞​x2+xx2−2x​.

Flashcard 32: What is the horizontal asymptote of f(x)=8x3+14x3+2f(x) = \frac{8x^3 + 1}{4x^3 + 2}f(x)=4x3+28x3+1​?

Flashcard 33: What is the horizontal asymptote of f(x)=x+4x2+2xf(x) = \frac{x + 4}{x^2 + 2x}f(x)=x2+2xx+4​?

Flashcard 34: Find the limit as xxx approaches infinity of f(x)=3x4x4+5f(x) = \frac{3x^4}{x^4 + 5}f(x)=x4+53x4​.

Flashcard 35: What is the horizontal asymptote of f(x)=8x3+14x3+2f(x) = \frac{8x^3 + 1}{4x^3 + 2}f(x)=4x3+28x3+1​?

Flashcard 36: Determine limx→−∞6x−12x+3\text{lim}_{x \to -\infty} \frac{6x - 1}{2x + 3}limx→−∞​2x+36x−1​.

Flashcard 37: Determine limx→∞2x3−53x3+1\text{lim}_{x \to \infty} \frac{2x^3 - 5}{3x^3 + 1}limx→∞​3x3+12x3−5​.

Flashcard 38: What is the horizontal asymptote of f(x)=4x3+5x3+6f(x) = \frac{4x^3 + 5}{x^3 + 6}f(x)=x3+64x3+5​?

Flashcard 39: What is the limit as xxx approaches infinity for f(x)=6x2−93x2+7f(x) = \frac{6x^2 - 9}{3x^2 + 7}f(x)=3x2+76x2−9​?

Flashcard 40: Find the horizontal asymptote of f(x)=3x+24x+5f(x) = \frac{3x + 2}{4x + 5}f(x)=4x+53x+2​.

Flashcard 41: What is the horizontal asymptote of f(x)=2x3+15x3+2f(x) = \frac{2x^3 + 1}{5x^3 + 2}f(x)=5x3+22x3+1​?

Flashcard 42: Find the limit as xxx approaches infinity of f(x)=4x22x2+3f(x) = \frac{4x^2}{2x^2 + 3}f(x)=2x2+34x2​.

Flashcard 43: State the horizontal asymptote of f(x)=7x2−x3x2+2f(x) = \frac{7x^2 - x}{3x^2 + 2}f(x)=3x2+27x2−x​.

Flashcard 44: What is the horizontal asymptote for f(x)=9x+73x+1f(x) = \frac{9x + 7}{3x + 1}f(x)=3x+19x+7​?

Flashcard 45: What is the limit as xxx approaches infinity for f(x)=3x3+2x2+4f(x) = \frac{3x^3 + 2}{x^2 + 4}f(x)=x2+43x3+2​?

Flashcard 46: What is the limit as xxx approaches infinity for f(x)=6x2−93x2+7f(x) = \frac{6x^2 - 9}{3x^2 + 7}f(x)=3x2+76x2−9​?

Flashcard 47: State the horizontal asymptote of f(x)=5x23x2+2f(x) = \frac{5x^2}{3x^2 + 2}f(x)=3x2+25x2​.

Flashcard 48: Find the limit as xxx approaches infinity of f(x)=3x4x4+5f(x) = \frac{3x^4}{x^4 + 5}f(x)=x4+53x4​.

Flashcard 49: Determine limx→∞5x2+33x2−2\text{lim}_{x \to \infty} \frac{5x^2 + 3}{3x^2 - 2}limx→∞​3x2−25x2+3​.

Flashcard 50: State the horizontal asymptote of f(x)=x2+xx3+1f(x) = \frac{x^2 + x}{x^3 + 1}f(x)=x3+1x2+x​.

Flashcard 51: What is the horizontal asymptote for f(x)=4x2−3x2+5xf(x) = \frac{4x^2 - 3}{x^2 + 5x}f(x)=x2+5x4x2−3​?

Flashcard 52: Find the horizontal asymptote of f(x)=6x33x3+1f(x) = \frac{6x^3}{3x^3 + 1}f(x)=3x3+16x3​.

Flashcard 53: What is the horizontal asymptote of f(x)=2x2+3x2+5f(x) = \frac{2x^2 + 3}{x^2 + 5}f(x)=x2+52x2+3​?

Flashcard 54: Determine limx→−∞6x−12x+3\text{lim}_{x \to -\infty} \frac{6x - 1}{2x + 3}limx→−∞​2x+36x−1​.

Flashcard 55: What is the horizontal asymptote for f(x)=9x+73x+1f(x) = \frac{9x + 7}{3x + 1}f(x)=3x+19x+7​?

Flashcard 56: State the horizontal asymptote of f(x)=x2+xx3+1f(x) = \frac{x^2 + x}{x^3 + 1}f(x)=x3+1x2+x​.

Flashcard 57: Find the horizontal asymptote of f(x)=6x33x3+1f(x) = \frac{6x^3}{3x^3 + 1}f(x)=3x3+16x3​.

Flashcard 58: Determine limx→−∞x2−2xx2+x\text{lim}_{x \to -\infty} \frac{x^2 - 2x}{x^2 + x}limx→−∞​x2+xx2−2x​.

Flashcard 59: What is the horizontal asymptote of f(x)=7x2x+3f(x) = \frac{7x}{2x + 3}f(x)=2x+37x​?

Flashcard 60: What is the horizontal asymptote for f(x)=2x2+3x5x2+1f(x) = \frac{2x^2 + 3x}{5x^2 + 1}f(x)=5x2+12x2+3x​?

Flashcard 61: What is the horizontal asymptote for f(x)=2x35x3−4f(x) = \frac{2x^3}{5x^3 - 4}f(x)=5x3−42x3​?

Flashcard 62: Find the limit as xxx approaches infinity of f(x)=4x22x2+3f(x) = \frac{4x^2}{2x^2 + 3}f(x)=2x2+34x2​.

Flashcard 63: Determine limx→∞5x−12x+3\text{lim}_{x \to \infty} \frac{5x - 1}{2x + 3}limx→∞​2x+35x−1​.

Flashcard 64: What is the limit as xxx approaches infinity for f(x)=2x+13x+4f(x) = \frac{2x + 1}{3x + 4}f(x)=3x+42x+1​?

Flashcard 65: Determine limx→∞7x2−52x2+4\text{lim}_{x \to \infty} \frac{7x^2 - 5}{2x^2 + 4}limx→∞​2x2+47x2−5​.

Flashcard 66: Determine lim⁡x→∞5x−12x+3\lim_{x \to \infty} \frac{5x - 1}{2x + 3}limx→∞​2x+35x−1​.

Flashcard 67: Find the horizontal asymptote of f(x)=x+1x2+1f(x) = \frac{x + 1}{x^2 + 1}f(x)=x2+1x+1​.

Flashcard 68: What is the horizontal asymptote for f(x)=4xx+5f(x) = \frac{4x}{x + 5}f(x)=x+54x​?

Flashcard 69: What is the horizontal asymptote of f(x)=x3−1x2+3xf(x) = \frac{x^3 - 1}{x^2 + 3x}f(x)=x2+3xx3−1​?

Flashcard 70: State the horizontal asymptote of f(x)=5xx3+2f(x) = \frac{5x}{x^3 + 2}f(x)=x3+25x​.

Flashcard 71: State the horizontal asymptote of f(x)=5x23x2+2f(x) = \frac{5x^2}{3x^2 + 2}f(x)=3x2+25x2​.

Flashcard 72: Determine limx→−∞−x3+43x3−x\text{lim}_{x \to -\infty} \frac{-x^3 + 4}{3x^3 - x}limx→−∞​3x3−x−x3+4​.

Flashcard 73: What is the horizontal asymptote of f(x)=4x3+5x3+6f(x) = \frac{4x^3 + 5}{x^3 + 6}f(x)=x3+64x3+5​?

Flashcard 74: What is the horizontal asymptote for f(x)=4x2−3x2+5xf(x) = \frac{4x^2 - 3}{x^2 + 5x}f(x)=x2+5x4x2−3​?

Flashcard 75: State the horizontal asymptote of f(x)=8xx2+3f(x) = \frac{8x}{x^2 + 3}f(x)=x2+38x​.

Flashcard 1: What is the limit as $x$ approaches infinity for $f(x) = \frac{2x + 1}{3x + 4}$ ?

Flashcard 2: Find the horizontal asymptote of $f(x) = \frac{3x + 2}{4x + 5}$ .

Flashcard 3: Find the limit as $x$ approaches infinity for $f(x) = \frac{7x^3}{2x^3 + 5}$ .

Flashcard 4: State the horizontal asymptote of $f(x) = \frac{5x}{x^3 + 2}$ .

Flashcard 5: Determine $\text{lim}_{x \to \infty} \frac{5x^2 + 3}{3x^2 - 2}$ .

Flashcard 6: Determine $\text{lim}_{x \to \infty} \frac{8x - 3}{2x + 1}$ .

Flashcard 7: What is the horizontal asymptote of $f(x) = \frac{2x^3 + 1}{5x^3 + 2}$ ?

Flashcard 8: What is the horizontal asymptote of $f(x) = \frac{3x + 2}{2x^2 + 5}$ ?

Flashcard 9: What is the limit as $x$ approaches $\infty$ for $f(x) = \frac{3x^3 + 2}{x^2 + 4}$ ?

Flashcard 10: What is the horizontal asymptote of $f(x) = \frac{x^3 - 1}{x^2 + 3x}$ ?

Flashcard 11: What is the horizontal asymptote of $f(x) = \frac{x^2}{x^3 + 1}$ ?

Flashcard 12: Determine $\text{lim}_{x \to -\infty} \frac{-x^3 + 4}{3x^3 - x}$ .

Flashcard 13: Determine $\text{lim}_{x \to \infty} \frac{8x - 3}{2x + 1}$ .

Flashcard 14: What is the horizontal asymptote of $f(x) = \frac{3x + 2}{2x^2 + 5}$ ?

Flashcard 15: State the horizontal asymptote of $f(x) = \frac{x^2 - 3}{2x^2 + x + 1}$ .

Flashcard 16: What is the horizontal asymptote of $f(x) = \frac{7x}{2x + 3}$ ?

Flashcard 17: State the horizontal asymptote of $f(x) = \frac{7x^2 - x}{3x^2 + 2}$ .

Flashcard 18: What is the horizontal asymptote for $f(x) = \frac{2x^3}{5x^3 - 4}$ ?

Flashcard 19: Find the limit as $x$ approaches infinity for $f(x) = \frac{7x^3}{2x^3 + 5}$ .

Flashcard 20: Find the limit as $x$ approaches infinity for $f(x) = \frac{x^2 - 1}{2x^2 + x}$ .

Flashcard 21: Determine $\text{lim}_{x \to \infty} \frac{2x^3 - 5}{3x^3 + 1}$ .

Flashcard 22: Determine $\text{lim}_{x \to \infty} \frac{7x^2 - 5}{2x^2 + 4}$ .

Flashcard 23: What is the horizontal asymptote of $f(x) = \frac{x^2}{x^3 + 1}$ ?

Flashcard 24: What is the horizontal asymptote for $f(x) = \frac{4x}{x + 5}$ ?

Flashcard 25: State the horizontal asymptote of $f(x) = \frac{8x}{x^2 + 3}$ .

Flashcard 26: Find the limit as $x$ approaches infinity for $f(x) = \frac{x^2 - 1}{2x^2 + x}$ .

Flashcard 27: Find the horizontal asymptote of $f(x) = \frac{x + 1}{x^2 + 1}$ .

Flashcard 28: What is the horizontal asymptote of $f(x) = \frac{2x^2 + 3}{x^2 + 5}$ ?

Flashcard 29: State the horizontal asymptote of $f(x) = \frac{x^2 - 3}{2x^2 + x + 1}$ .

Flashcard 30: What is the horizontal asymptote for $f(x) = \frac{2x^2 + 3x}{5x^2 + 1}$ ?

Flashcard 31: Determine $\text{lim}_{x \to -\infty} \frac{x^2 - 2x}{x^2 + x}$ .

Flashcard 32: What is the horizontal asymptote of $f(x) = \frac{8x^3 + 1}{4x^3 + 2}$ ?

Flashcard 33: What is the horizontal asymptote of $f(x) = \frac{x + 4}{x^2 + 2x}$ ?

Flashcard 34: Find the limit as $x$ approaches infinity of $f(x) = \frac{3x^4}{x^4 + 5}$ .

Flashcard 35: What is the horizontal asymptote of $f(x) = \frac{8x^3 + 1}{4x^3 + 2}$ ?

Flashcard 36: Determine $\text{lim}_{x \to -\infty} \frac{6x - 1}{2x + 3}$ .

Flashcard 37: Determine $\text{lim}_{x \to \infty} \frac{2x^3 - 5}{3x^3 + 1}$ .

Flashcard 38: What is the horizontal asymptote of $f(x) = \frac{4x^3 + 5}{x^3 + 6}$ ?

Flashcard 39: What is the limit as $x$ approaches infinity for $f(x) = \frac{6x^2 - 9}{3x^2 + 7}$ ?

Flashcard 40: Find the horizontal asymptote of $f(x) = \frac{3x + 2}{4x + 5}$ .

Flashcard 41: What is the horizontal asymptote of $f(x) = \frac{2x^3 + 1}{5x^3 + 2}$ ?

Flashcard 42: Find the limit as $x$ approaches infinity of $f(x) = \frac{4x^2}{2x^2 + 3}$ .

Flashcard 43: State the horizontal asymptote of $f(x) = \frac{7x^2 - x}{3x^2 + 2}$ .

Flashcard 44: What is the horizontal asymptote for $f(x) = \frac{9x + 7}{3x + 1}$ ?

Flashcard 45: What is the limit as $x$ approaches infinity for $f(x) = \frac{3x^3 + 2}{x^2 + 4}$ ?

Flashcard 46: What is the limit as $x$ approaches infinity for $f(x) = \frac{6x^2 - 9}{3x^2 + 7}$ ?

Flashcard 47: State the horizontal asymptote of $f(x) = \frac{5x^2}{3x^2 + 2}$ .

Flashcard 48: Find the limit as $x$ approaches infinity of $f(x) = \frac{3x^4}{x^4 + 5}$ .

Flashcard 49: Determine $\text{lim}_{x \to \infty} \frac{5x^2 + 3}{3x^2 - 2}$ .

Flashcard 50: State the horizontal asymptote of $f(x) = \frac{x^2 + x}{x^3 + 1}$ .

Flashcard 51: What is the horizontal asymptote for $f(x) = \frac{4x^2 - 3}{x^2 + 5x}$ ?

Flashcard 52: Find the horizontal asymptote of $f(x) = \frac{6x^3}{3x^3 + 1}$ .

Flashcard 53: What is the horizontal asymptote of $f(x) = \frac{2x^2 + 3}{x^2 + 5}$ ?

Flashcard 54: Determine $\text{lim}_{x \to -\infty} \frac{6x - 1}{2x + 3}$ .

Flashcard 55: What is the horizontal asymptote for $f(x) = \frac{9x + 7}{3x + 1}$ ?

Flashcard 56: State the horizontal asymptote of $f(x) = \frac{x^2 + x}{x^3 + 1}$ .

Flashcard 57: Find the horizontal asymptote of $f(x) = \frac{6x^3}{3x^3 + 1}$ .

Flashcard 58: Determine $\text{lim}_{x \to -\infty} \frac{x^2 - 2x}{x^2 + x}$ .

Flashcard 59: What is the horizontal asymptote of $f(x) = \frac{7x}{2x + 3}$ ?

Flashcard 60: What is the horizontal asymptote for $f(x) = \frac{2x^2 + 3x}{5x^2 + 1}$ ?

Flashcard 61: What is the horizontal asymptote for $f(x) = \frac{2x^3}{5x^3 - 4}$ ?

Flashcard 62: Find the limit as $x$ approaches infinity of $f(x) = \frac{4x^2}{2x^2 + 3}$ .

Flashcard 63: Determine $\text{lim}_{x \to \infty} \frac{5x - 1}{2x + 3}$ .

Flashcard 64: What is the limit as $x$ approaches infinity for $f(x) = \frac{2x + 1}{3x + 4}$ ?

Flashcard 65: Determine $\text{lim}_{x \to \infty} \frac{7x^2 - 5}{2x^2 + 4}$ .

Flashcard 66: Determine $\lim_{x \to \infty} \frac{5x - 1}{2x + 3}$ .

Flashcard 67: Find the horizontal asymptote of $f(x) = \frac{x + 1}{x^2 + 1}$ .

Flashcard 68: What is the horizontal asymptote for $f(x) = \frac{4x}{x + 5}$ ?

Flashcard 69: What is the horizontal asymptote of $f(x) = \frac{x^3 - 1}{x^2 + 3x}$ ?

Flashcard 70: State the horizontal asymptote of $f(x) = \frac{5x}{x^3 + 2}$ .

Flashcard 71: State the horizontal asymptote of $f(x) = \frac{5x^2}{3x^2 + 2}$ .

Flashcard 72: Determine $\text{lim}_{x \to -\infty} \frac{-x^3 + 4}{3x^3 - x}$ .

Flashcard 73: What is the horizontal asymptote of $f(x) = \frac{4x^3 + 5}{x^3 + 6}$ ?

Flashcard 74: What is the horizontal asymptote for $f(x) = \frac{4x^2 - 3}{x^2 + 5x}$ ?

Flashcard 75: State the horizontal asymptote of $f(x) = \frac{8x}{x^2 + 3}$ .

Flashcard 76: What is the horizontal asymptote of $f(x) = \frac{x + 4}{x^2 + 2x}$ ?

Flashcard 77: Find the horizontal asymptote of $f(x) = \frac{5x^3 - 2x}{4x^3 + x}$ .

Flashcard 78: Find the horizontal asymptote of $f(x) = \frac{5x^3 - 2x}{4x^3 + x}$ .