Infinite Discontinuity

Definition

An infinite discontinuity occurs when a function has a vertical asymptote, meaning the function approaches positive or negative infinity at a particular point. This type of discontinuity is considered 'infinite' because the function's value cannot be defined at that point, as it would result in division by zero, which is undefined.

5 Must Know Facts For Your Next Test

1. Infinite discontinuities occur when the denominator of a rational function is equal to zero, causing the function to approach positive or negative infinity.
2. Functions with infinite discontinuities typically have vertical asymptotes at the points of discontinuity, which the graph of the function approaches but never touches.
3. Identifying and understanding infinite discontinuities is crucial for sketching the graph of a function and analyzing its behavior.
4. Infinite discontinuities can be detected by setting the denominator of a rational function equal to zero and solving for the x-value(s) that result in a division by zero.
5. Unlike removable discontinuities, infinite discontinuities cannot be 'fixed' by redefining the function at the point of discontinuity.

Review Questions

• Explain the concept of an infinite discontinuity and how it relates to the behavior of a function's graph.
  • An infinite discontinuity occurs when a function has a vertical asymptote, meaning the function approaches positive or negative infinity at a particular point. This type of discontinuity is considered 'infinite' because the function's value cannot be defined at that point, as it would result in division by zero, which is undefined. The function's graph will approach the vertical asymptote but never touch it, indicating the infinite discontinuity.
• Describe the process of identifying an infinite discontinuity in a rational function.
  • To identify an infinite discontinuity in a rational function, you need to set the denominator of the function equal to zero and solve for the x-value(s) that result in a division by zero. These x-values will correspond to the points of infinite discontinuity, where the function has a vertical asymptote. Unlike removable discontinuities, infinite discontinuities cannot be 'fixed' by redefining the function at the point of discontinuity.
• Analyze the significance of understanding infinite discontinuities in the context of sketching the graph of a function and analyzing its behavior.
  • Identifying and understanding infinite discontinuities is crucial for accurately sketching the graph of a function and analyzing its behavior. Knowing the location of the vertical asymptotes, which correspond to the points of infinite discontinuity, allows you to determine the overall shape and characteristics of the function's graph. This information is essential for making predictions about the function's behavior, such as its rate of change, intervals of increase or decrease, and any asymptotic tendencies. Mastering the concept of infinite discontinuity is a key skill for success in topics related to continuity and function analysis.

Related terms

Vertical Asymptote:

A vertical line that a function's graph approaches but never touches, indicating where the function has an infinite discontinuity.

Removable Discontinuity:

A discontinuity that can be 'removed' by redefining the function at the point of discontinuity, unlike an infinite discontinuity which cannot be removed.

Point of Discontinuity:

The specific point where a function is not defined or has a jump in its value, such as the point of an infinite discontinuity.