Infinite limit

Definition

An infinite limit refers to a situation where the values of a function increase or decrease without bound as the input approaches a certain point. This concept is crucial in understanding behavior near vertical asymptotes or the unbounded growth of functions, which can be observed when limits are evaluated at specific points or as inputs approach infinity.

5 Must Know Facts For Your Next Test

1. Infinite limits can occur either as the input approaches a finite number or as it approaches positive or negative infinity.
2. When evaluating an infinite limit at a specific point, it often indicates that the function has a vertical asymptote at that point.
3. A function can have different infinite limits depending on whether the input approaches from the left or the right, leading to one-sided limits.
4. In cases of limits at infinity, if the degree of the numerator is greater than the degree of the denominator in a rational function, the limit will be infinite.
5. Understanding infinite limits is essential for analyzing the behavior of functions and solving problems related to vertical asymptotes and end behavior.

Review Questions

• How do infinite limits relate to vertical asymptotes in a function's graph?
  • Infinite limits are directly related to vertical asymptotes since they occur when a function's value tends toward infinity as it approaches a certain input. When evaluating a limit and finding that it is infinite at a specific point, this typically indicates that there is a vertical asymptote present on the graph at that point. The function will grow without bound as it nears this vertical line.
• Discuss how one-sided limits can affect the determination of an infinite limit.
  • One-sided limits are important when determining an infinite limit because they help identify how a function behaves as it approaches a point from either side. If the left-hand limit and right-hand limit both approach infinity but potentially differ in direction (positive or negative), this can indicate differing behaviors of the function near vertical asymptotes. Understanding these one-sided limits provides clarity on whether an infinite limit exists and informs us about the nature of discontinuities.
• Evaluate the significance of infinite limits when analyzing rational functions and their end behavior.
  • Infinite limits are crucial for analyzing rational functions because they reveal how these functions behave as their inputs grow very large or very small. For instance, if we look at rational functions where the degree of the numerator exceeds that of the denominator, we find that the function tends toward infinity as x approaches either positive or negative infinity. This behavior helps us understand the end behavior of functions, which is essential for sketching graphs and identifying trends in data.

Related terms

Vertical Asymptote: A vertical asymptote is a line that a graph approaches but never touches, typically occurring when a function's value becomes infinitely large as it approaches a certain input.

Limit at Infinity: A limit at infinity describes the behavior of a function as the input grows larger and larger, determining whether the function approaches a finite value, infinite value, or diverges.

Continuous Function:

A continuous function is one that does not have any breaks, jumps, or holes in its graph, meaning that small changes in the input result in small changes in the output.