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Asymptote Calculator — Vertical, Horizontal, Slant Asymptotes and Holes

Q: How do you find vertical asymptotes?

Factor and cancel any common factors, then set the remaining denominator equal to zero. Each real solution is a vertical asymptote x = a.

Q: How do you find the horizontal asymptote?

Compare degrees: if the numerator’s degree is smaller, y = 0; if they are equal, y = ratio of the leading coefficients; if the numerator is larger, there is no horizontal asymptote.

Q: When is there a slant (oblique) asymptote?

When the numerator’s degree is exactly one more than the denominator’s. Divide P by Q; the quotient (without the remainder) is the slant asymptote y = mx + b.

Q: What is the difference between a hole and a vertical asymptote?

A hole comes from a factor that cancels — the graph has a single missing point. A vertical asymptote comes from a zero of the denominator that does not cancel — the graph shoots off to ±∞.

Q: Can a graph cross its horizontal asymptote?

Yes. A horizontal asymptote describes behavior as x → ±∞; the graph may cross it at finite x. It can never cross a vertical asymptote.

Find every asymptote of a rational function P(x)/Q(x): vertical asymptotes, the horizontal or slant (oblique) asymptote, holes, the domain and the x- and y-intercepts.

Numerator P(x)

Denominator Q(x)

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How it works

Cancel common factors of P and Q (their gcd) → holes at the cancelled zeros · vertical asymptotes: zeros of the reduced denominator · degree n of P vs m of Q: n < m → y = 0; n = m → y = (leading P)/(leading Q); n = m + 1 → slant asymptote = quotient of P ÷ Q

A rational function has a vertical asymptote where its reduced denominator is zero, and a hole where a factor cancels between numerator and denominator. Its end behavior is set by the degrees: a lower-degree numerator gives the horizontal asymptote y = 0, equal degrees give the ratio of leading coefficients, and a numerator one degree higher gives a slant asymptote found by polynomial long division (OpenStax College Algebra §5.6). The calculator cancels the greatest common factor exactly and finds the zeros numerically.

Worked example

f(x) = (x² − 4)/(x² − x − 2) = (x − 2)(x + 2)/[(x − 2)(x + 1)]. The common factor x − 2 cancels, leaving (x + 2)/(x + 1) with a hole at x = 2, y = 4/3. The reduced denominator is zero at x = −1, so the vertical asymptote is x = −1. Both degrees are 2 with leading coefficients 1 and 1, so the horizontal asymptote is y = 1. Domain: all real numbers except x = −1, 2; x-intercept (−2, 0); y-intercept (0, 2).

Sources: OpenStax College Algebra 2e — 5.6 Rational Functions · Paul’s Online Math Notes — Rational Functions

Frequently asked questions

How do you find vertical asymptotes?

Factor and cancel any common factors, then set the remaining denominator equal to zero. Each real solution is a vertical asymptote x = a.

How do you find the horizontal asymptote?

Compare degrees: if the numerator’s degree is smaller, y = 0; if they are equal, y = ratio of the leading coefficients; if the numerator is larger, there is no horizontal asymptote.

When is there a slant (oblique) asymptote?

When the numerator’s degree is exactly one more than the denominator’s. Divide P by Q; the quotient (without the remainder) is the slant asymptote y = mx + b.

What is the difference between a hole and a vertical asymptote?

A hole comes from a factor that cancels — the graph has a single missing point. A vertical asymptote comes from a zero of the denominator that does not cancel — the graph shoots off to ±∞.

Can a graph cross its horizontal asymptote?

Yes. A horizontal asymptote describes behavior as x → ±∞; the graph may cross it at finite x. It can never cross a vertical asymptote.

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This calculator is for educational purposes. Double-check important results before acting on them.