Understanding how to find holes in a function is an essential skill for anyone studying algebra, precalculus, or calculus. A hole—also known as a removable discontinuity—occurs when a function is undefined at a single point, yet the graph approaches that point from both sides without breaking into infinity. Unlike vertical asymptotes, holes represent missing points that can be "filled in" to make the function continuous. Learning to locate these gaps helps students analyze rational functions, sketch accurate graphs, and evaluate limits with confidence Easy to understand, harder to ignore..
What Is a Hole in a Function?
In mathematics, a hole is a specific type of discontinuity where a function has no defined value at a particular input, but the surrounding values behave predictably. This most commonly appears in rational functions, which are fractions where both the numerator and denominator are polynomials.
A hole exists at $x = c$ when:
- The original function is undefined at $x = c$ because the denominator equals zero.
- The numerator also equals zero at $x = c$, creating a $0/0$ indeterminate form.
- After factoring and canceling the common term, the simplified function is defined at $
