Infinite discontinuity

Definition

An infinite discontinuity occurs at a point where the function approaches infinity or negative infinity as the input approaches a certain value. This results in an unbounded behavior of the function at that specific point.

5 Must Know Facts For Your Next Test

1. Infinite discontinuities are characterized by vertical asymptotes.
2. The limit of the function does not exist at points of infinite discontinuity.
3. Graphically, the function will show a 'break' with values shooting off to positive or negative infinity near the discontinuity.
4. Infinite discontinuities often occur in rational functions where the denominator is zero and cannot be canceled out.
5. Limits approaching from the left and right can both approach infinity, negative infinity, or differ, indicating an infinite discontinuity.

Review Questions

• What happens to a function's value as it approaches an infinite discontinuity?
• How can you identify an infinite discontinuity on a graph?
• Why do limits not exist at points of infinite discontinuity?

Related terms

Vertical Asymptote: A line $x = a$ where the function grows without bound as $x$ approaches $a$.

Removable Discontinuity: A point on the graph that can be 'fixed' by redefining the function at that point.

Jump Discontinuity: $\text{A point where } \lim_{{x \to c^-}} f(x) \neq \lim_{{x \to c^+}} f(x) \text{ but both are finite.}$