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Hole (in a Graph)

A hole in a graph is a point where the function is undefined, even though the graph exists on both sides of that point. It appears as a small open circle on the graph, indicating a single missing value.

A hole occurs at a point x=ax = a when a factor cancels from both the numerator and denominator of a rational function, leaving the function undefined at x=ax = a despite having a finite limit there. Formally, lim⁡x→af(x)\lim_{x \to a} f(x) exists, but f(a)f(a) is undefined. The hole is plotted at the coordinates (a,L)(a, L), where LL is the value of the limit. This type of discontinuity is also called a removable discontinuity because the function could be redefined at that point to fill the gap.

Worked Example

Problem: Find the hole in the graph of f(x)=x2−4x−2f(x) = \dfrac{x^2 - 4}{x - 2}.
Factor the numerator: The numerator is a difference of squares, so factor it.
f(x)=(x−2)(x+2)x−2f(x) = \frac{(x - 2)(x + 2)}{x - 2}
Identify the common factor: Both the numerator and denominator contain the factor (x−2)(x - 2). This factor equals zero when x=2x = 2, so the function is undefined at x=2x = 2.
Cancel the common factor: After canceling, the simplified function is a line — but remember, the original function is still undefined at x=2x = 2.
f(x)=x+2,x≠2f(x) = x + 2, \quad x \neq 2
Find the y-coordinate of the hole: Substitute x=2x = 2 into the simplified expression to find where the hole sits on the graph.
y=2+2=4y = 2 + 2 = 4
State the hole: The graph has a hole at the point (2,4)(2, 4). You would draw an open circle there to show the function is undefined at that exact location.
Answer: The graph of f(x)=x2−4x−2f(x) = \dfrac{x^2 - 4}{x - 2} has a hole at the point (2,4)(2, 4).

Visualization

Why It Matters

Holes come up frequently when you work with rational functions in precalculus and calculus. Recognizing them is essential for sketching accurate graphs and for understanding limits, since a function can approach a value at a hole without ever reaching it. In calculus, the entire concept of a derivative relies on evaluating limits at points that initially look like holes.

Common Mistakes

Mistake: Confusing a hole with a vertical asymptote
Correction: A hole occurs when a factor cancels completely from both the numerator and denominator. A vertical asymptote occurs when a factor remains in the denominator after canceling. Always simplify the rational expression first to tell the difference.
Mistake: Forgetting to find the y-coordinate of the hole
Correction: A hole is a point, not just an x-value. After identifying the x-value that causes the cancellation, substitute it into the simplified function to get the y-coordinate.

Related Terms