Hole (in a Graph) — Definition, Formula & Examples

A hole in a graph is a single missing point where the function is undefined, but the graph continues smoothly on either side. It occurs when a factor cancels from both the numerator and denominator of a rational function.

A hole (removable discontinuity) exists at $x = a$ when $\lim_{x \to a} f(x) = L$ exists and is finite, but $f(a)$ is either undefined or not equal to $L$ . The point $(a, L)$ is absent from the graph.

How It Works

Holes appear in rational functions when a common factor divides out of the numerator and denominator. To find a hole, factor both the numerator and denominator completely, then cancel any shared factors. The

x

-value that made the cancelled factor zero is the hole's

x

-coordinate. Substitute that

x

-value into the simplified function to get the

y

-coordinate. On a graph, you represent the hole as an open circle at that point.

Worked Example

Problem: Find the hole in the graph of

f(x) = \dfrac{x^2 - 4}{x - 2}

Factor the numerator: Recognize the difference of squares in the numerator.

f(x) = \frac{(x - 2)(x + 2)}{x - 2}

Cancel the common factor: The factor

(x - 2)

appears in both numerator and denominator, so it cancels. The original function is still undefined at

x = 2

f(x) = x + 2, \quad x \neq 2

Find the y-coordinate: Substitute

x = 2

into the simplified expression to find the missing

y

-value.

y = 2 + 2 = 4

Answer: There is a hole at the point

(2, 4)

. The graph looks like the line

y = x + 2

with an open circle at

(2, 4)

Why It Matters

Identifying holes is essential in precalculus and calculus when sketching rational functions and evaluating limits. In calculus, removable discontinuities show up frequently when computing limits algebraically — recognizing a hole lets you evaluate a limit by cancellation instead of resorting to more advanced techniques.

Common Mistakes

Mistake: Confusing a hole with a vertical asymptote.

Correction: A hole occurs when a factor cancels completely from numerator and denominator. A vertical asymptote occurs when a factor remains in the denominator after cancellation. Always factor and simplify first before deciding which you have.