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Vertical Asymptotes

A free Precalculus lesson from the “Rational Functions” unit, with a worked example and practice problems including step-by-step solutions.

Vertical asymptotes mark x-values where the simplified denominator is zero. This lesson is part of Precalculus: Advanced Functions, so the emphasis is on interpreting behavior, choosing the right representation, and explaining the result clearly rather than memorizing isolated algebra moves.

What you'll learn

Why it matters: Rational functions model rates, constraints, efficiency, and quantities that change sharply near restricted inputs.

Worked example

Problem. Find the vertical asymptote x-value of r(x) = 3/(x - 3).

  1. A vertical asymptote occurs where the (uncanceled) denominator equals 0.
  2. Solve x - 3 = 0, so x = 3.
  3. No factor cancels, so the vertical asymptote is x = 3.

Answer: 3

Practice problems

1. Find the vertical asymptote x-value of r(x) = 3/(x - 3).

Show solution
  1. A vertical asymptote occurs where the (uncanceled) denominator equals 0.
  2. Solve x - 3 = 0, so x = 3.
  3. No factor cancels, so the vertical asymptote is x = 3.

Answer: 3

2. Find the vertical asymptote x-value of r(x) = 4/(x + 4).

Show solution
  1. Warm-up: First identify exactly what the question is asking: Find the vertical asymptote x-value of r(x) = 4/(x + 4).
  2. Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
  3. Set the denominator equal to 0: x + 4 = 0.
  4. Solve to get x = -4.
  5. So the vertical asymptote is x = -4.
  6. Check the result by substituting or estimating: the response should match -4 and make sense in the original problem.

Answer: -4

3. r(x) = 5/((x - 5)(x - 12)). Enter the LARGER of the two vertical-asymptote x-values.

Show solution
  1. Each denominator factor set to 0 gives a vertical asymptote.
  2. x - 5 = 0 gives x = 5; x - 12 = 0 gives x = 12.
  3. The larger value is x = 12.

Answer: 12

4. r(x) = 6/((x - 6)(x - 10)). Enter the SMALLER of the two vertical-asymptote x-values.

Show solution
  1. Core Practice: First identify exactly what the question is asking: r(x) = 6/((x - 6)(x - 10)). Enter the SMALLER of the two vertical-asymptote x-values.
  2. Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
  3. Set each factor of the denominator to 0.
  4. x = 6 and x = 10 are the two asymptotes.
  5. The smaller value is x = 6.
  6. Check the result by substituting or estimating: the response should match 6 and make sense in the original problem.

Answer: 6

5. How many vertical asymptotes does r(x) = 2/((x - 2)(x - 7)) have?

Show solution
  1. Count the distinct zeros of the uncanceled denominator.
  2. x = 2 and x = 7 are both zeros and neither cancels.
  3. That is 2 vertical asymptotes.

Answer: 2

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