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Rational Function Asymptote Calculator

Find vertical and horizontal asymptotes for (ax+b)/(cx+d)

R(x) = (x + ) / (x +)

Asymptote Rules

Vertical asymptote: denominator = 0 and not canceled

Horizontal asymptote for (ax+b)/(cx+d): y = a / c when c != 0

If a common factor cancels, the point is a hole instead of a vertical asymptote

Asymptotes describe graph behavior near restricted x-values and far from the origin. For linear-over-linear rational functions, the horizontal asymptote comes from the ratio of leading coefficients.

Vertical Asymptote

Occurs where the denominator approaches zero and the numerator does not cancel the factor.

Horizontal Asymptote

Shows the value approached by the function as x becomes very large in magnitude.

Hole Check

A shared numerator and denominator root creates a removable discontinuity, not a vertical asymptote.

Graphing Use

Asymptotes act as guide lines for sketching rational function graphs.

Teaching Example: for (2x+1)/(3x-6), the vertical asymptote is x=2 and the horizontal asymptote is y=2/3.

Frequently Asked Questions

How do you find a vertical asymptote?▼

Set the denominator equal to zero and check that the factor does not cancel with the numerator.

How do you find a horizontal asymptote?▼

For equal degrees, divide the leading coefficients. For lower numerator degree, the horizontal asymptote is y=0.

Can a hole replace an asymptote?▼

Yes. If the denominator zero also cancels with a numerator factor, the graph has a removable hole instead of a vertical asymptote.

What does end behavior mean?▼

End behavior describes what the function approaches as x goes toward positive or negative infinity.

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