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Rational Function Hole Calculator

Detect removable discontinuities from canceled factors

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

Hole Rule

Factor numerator and denominator

Common factor cancels -> removable discontinuity

Hole x-value = zero of canceled factor

Hole y-value = simplified function evaluated at that x

A hole is not visible from the simplified formula alone. It records an excluded point from the original rational function after a factor cancels.

Common Root

For linear factors, a hole exists when numerator and denominator have the same root.

Original Domain

The hole x-value is still excluded from the original function domain.

Graph Meaning

The graph follows the simplified function but has an open circle at the hole.

Limit Connection

The limit at a removable discontinuity usually equals the hole y-value.

Teaching Example: (2x-4)/(3x-6) = 2(x-2)/3(x-2). The factor x-2 cancels, so the graph has a hole at x=2 with y=2/3.

Frequently Asked Questions

What is a hole in a rational function?▼

A hole is a removable discontinuity created when a numerator factor cancels with a denominator factor.

How do you find a hole?▼

Factor numerator and denominator, cancel common factors, then use the canceled factor zero as the hole x-value.

How do you find the y-value of a hole?▼

Substitute the hole x-value into the simplified function after canceling the common factor.

Is a hole the same as a vertical asymptote?▼

No. A vertical asymptote remains after simplification, while a hole comes from a canceled factor.

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