Quick Answer

For f(x) = (x + 1)/(x^2 - 4), the vertical asymptotes are x = 2 and x = -2 (roots of x^2 - 4), and the horizontal asymptote is y = 0 (since the numerator's degree is less than the denominator's).

Numerator P(x)

Degree of P(x) 0 (constant) 1 (linear) 2 (quadratic) 3 (cubic) 4 (quartic) 5 (quintic)

Denominator Q(x)

Degree of Q(x) 0 (constant) 1 (linear) 2 (quadratic) 3 (cubic) 4 (quartic) 5 (quintic)

Results

Common Examples

Input	Result
f(x) = 1/(x - 3)	Vertical: x = 3, Horizontal: y = 0
f(x) = (2x + 1)/(x - 1)	Vertical: x = 1, Horizontal: y = 2
f(x) = (x^2 + 1)/(x - 2)	Vertical: x = 2, Oblique: y = x + 2
f(x) = x/(x^2 - 9)	Vertical: x = 3 and x = -3, Horizontal: y = 0

How It Works

For a rational function f(x) = P(x) / Q(x), where P(x) and Q(x) are polynomials, there are three types of asymptotes:

Vertical Asymptotes

Vertical asymptotes occur at x-values where the denominator equals zero and the numerator does not equal zero. If both numerator and denominator share a common root, that root creates a hole (removable discontinuity) instead of an asymptote. To find vertical asymptotes, solve Q(x) = 0 and exclude any roots that are also roots of P(x).

Horizontal Asymptotes

The horizontal asymptote depends on the degrees of P(x) and Q(x):

• If deg(P) < deg(Q), the horizontal asymptote is y = 0
• If deg(P) = deg(Q), the horizontal asymptote is y = (leading coefficient of P) / (leading coefficient of Q)
• If deg(P) > deg(Q), there is no horizontal asymptote

Oblique (Slant) Asymptotes

An oblique asymptote exists when the degree of the numerator is exactly one greater than the degree of the denominator. To find it, perform polynomial long division of P(x) by Q(x). The quotient (ignoring the remainder) gives the equation y = mx + b.

Worked Example

For f(x) = (x^2 + 1) / (x - 2): The denominator x - 2 = 0 gives a vertical asymptote at x = 2. The numerator has degree 2 and the denominator has degree 1, so deg(P) = deg(Q) + 1, meaning there is an oblique asymptote. Dividing x^2 + 1 by x - 2: x^2 / x = x, then x(x - 2) = x^2 - 2x. Subtracting gives 2x + 1. Then 2x / x = 2, and 2(x - 2) = 2x - 4. Subtracting gives remainder 5. The oblique asymptote is y = x + 2.

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Frequently Asked Questions

What is an asymptote?

An asymptote is a line that a function's graph approaches as x or y tends toward infinity or a specific value. The graph gets arbitrarily close to the asymptote but typically does not cross it (though horizontal and oblique asymptotes can be crossed at finite values). Asymptotes describe the end behavior and discontinuities of a function.

Can a function have both a horizontal and an oblique asymptote?

No. A rational function can have at most one of either a horizontal asymptote or an oblique asymptote, never both. If the numerator and denominator have the same degree, there is a horizontal asymptote. If the numerator's degree is exactly one higher, there is an oblique asymptote. If the numerator's degree is more than one higher, there is neither.

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

Both occur where the denominator equals zero. A hole (removable discontinuity) happens when a factor cancels between the numerator and denominator. A vertical asymptote happens when the factor appears only in the denominator. For example, in (x-1)(x+2)/((x-1)(x-3)), x = 1 is a hole because (x-1) cancels, and x = 3 is a vertical asymptote.

How do I enter polynomial coefficients?

Select the degree of the polynomial, then enter the coefficient for each term. For example, for x^2 - 4, select degree 2 and enter 1 for the x^2 coefficient, 0 for the x coefficient, and -4 for the constant. All unset coefficients default to 0.

What are the limitations of this calculator?

This calculator finds asymptotes for rational functions (polynomial divided by polynomial). It handles polynomials up to degree 5. For root-finding on higher-degree polynomials, it uses the rational root theorem, which works best with integer and rational roots. Irrational roots of degree 3+ polynomials may not always be found.