Limit Graph

Definition

A limit graph visually represents the behavior of a function as it approaches a certain value or infinity. It helps to illustrate concepts such as infinite limits, which occur when the function approaches an unbounded value, and limits at infinity, where the input values grow without bound. By observing the limit graph, one can determine the behavior of a function around vertical and horizontal asymptotes, revealing essential information about continuity and discontinuity.

5 Must Know Facts For Your Next Test

1. Limit graphs are particularly useful for visualizing how functions behave near points of discontinuity and at infinity.
2. An infinite limit is indicated on a limit graph by arrows extending vertically upwards or downwards, signifying that the function grows without bounds.
3. Horizontal asymptotes on a limit graph show the behavior of functions as the input approaches positive or negative infinity.
4. The shape of the limit graph can reveal whether a function has removable or non-removable discontinuities.
5. Limit graphs can help predict the output values of functions even if they are undefined at certain points due to discontinuities.

Review Questions

• How does a limit graph illustrate the concept of infinite limits and what specific features would you look for?
  • A limit graph illustrates infinite limits by showing how a function's values increase or decrease indefinitely as they approach certain points. Specific features to look for include vertical arrows indicating unbounded growth and the presence of vertical asymptotes where the function is undefined. By examining these aspects, one can understand how the function behaves near points of discontinuity and what its output tends toward.
• Discuss how horizontal asymptotes in a limit graph relate to limits at infinity.
  • Horizontal asymptotes in a limit graph indicate the value that a function approaches as its input values grow infinitely large or small. When evaluating limits at infinity, if a function settles at a constant value as the input approaches positive or negative infinity, this creates a horizontal asymptote. Therefore, analyzing these asymptotes helps identify the long-term behavior of functions beyond their immediate surroundings.
• Evaluate how understanding limit graphs enhances your ability to analyze continuity and discontinuity in functions.
  • Understanding limit graphs enhances your ability to analyze continuity and discontinuity by providing a visual representation of where functions may not behave as expected. For instance, if a limit graph shows gaps or vertical asymptotes, it indicates points where continuity fails. This allows for deeper analysis of how functions change at specific intervals, leading to better predictions of their overall behavior and helping to solve complex calculus problems involving limits.

Related terms

Asymptote: A line that a graph approaches but never touches, often representing infinite limits or behavior at infinity.

Continuity: A property of a function that indicates there are no breaks, jumps, or holes in the graph, allowing for limits to be evaluated smoothly.

Infinite Limit: Occurs when the values of a function increase or decrease without bound as the input approaches a certain point.