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Mathematics, five units · Ages 17–18

Finding Asymptotes of Rational Functions Worksheet

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Designed for 12th-grade mathematics students. Determine all vertical asymptotes, holes, and horizontal asymptotes for each rational function. Simplify and verify factors carefully.

  1. Find all asymptotes of f(x) = 1 / (x - 4).

  2. Find all asymptotes of f(x) = (2x + 5) / (x - 1).

  3. Determine whether x = -3 is an asymptote or a hole for f(x) = (x + 3) / (x^2 - 9).

  4. Find the horizontal asymptote of f(x) = (5x^2 - 2) / (2x^2 + 3x + 1).

  5. Find all asymptotes of f(x) = (3x^2) / (x^2 - 16).

  6. Find all asymptotes of f(x) = (x^3 + 1) / (x^2 - 4).

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Finding Asymptotes of Rational Functions Worksheet

Answer key

  1. Find all asymptotes of f(x) = 1 / (x - 4).

    Vertical asymptote: x = 4. Horizontal asymptote: y = 0.

  2. Find all asymptotes of f(x) = (2x + 5) / (x - 1).

    Vertical asymptote: x = 1. Horizontal asymptote: y = 2.

  3. Determine whether x = -3 is an asymptote or a hole for f(x) = (x + 3) / (x^2 - 9).

    Hole at x = -3 (factor cancels to 1 / (x - 3)). Vertical asymptote is at x = 3.

  4. Find the horizontal asymptote of f(x) = (5x^2 - 2) / (2x^2 + 3x + 1).

    Horizontal asymptote: y = 5/2 (degrees are equal; ratio of leading coefficients).

  5. Find all asymptotes of f(x) = (3x^2) / (x^2 - 16).

    Vertical asymptotes: x = 4 and x = -4. Horizontal asymptote: y = 3.

  6. Find all asymptotes of f(x) = (x^3 + 1) / (x^2 - 4).

    Vertical asymptotes: x = 2 and x = -2. Horizontal asymptote: None (numerator degree is higher).