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📏honors pre-calculus review

key term - Infinite Limit

Citation:

Definition

An infinite limit refers to a mathematical concept where a function's values approach an unbounded or infinitely large value as the input variable approaches a specific point. This term is particularly relevant in the context of finding limits using numerical and graphical approaches.

5 Must Know Facts For Your Next Test

  1. An infinite limit occurs when a function's values approach positive or negative infinity as the input variable approaches a specific point.
  2. Infinite limits are often indicated by the symbols $\lim_{x \to a} f(x) = \infty$ or $\lim_{x \to a} f(x) = -\infty$.
  3. Vertical asymptotes on a function's graph represent points where the function has an infinite limit.
  4. Identifying infinite limits is crucial for understanding the behavior and properties of a function, such as its domain and range.
  5. Numerical and graphical approaches can be used to determine the presence and value of infinite limits for a given function.

Review Questions

  • Explain how the concept of an infinite limit relates to the behavior of a function's graph.
    • The presence of an infinite limit in a function indicates that the function's values approach positive or negative infinity as the input variable approaches a specific point. This is visually represented on the function's graph by the presence of a vertical asymptote, which is a vertical line that the graph approaches but never touches. The function's values will continue to increase or decrease without bound as the input variable approaches the value corresponding to the infinite limit.
  • Describe the steps involved in using numerical and graphical approaches to determine the presence and value of an infinite limit.
    • To determine the presence and value of an infinite limit using numerical and graphical approaches, one can follow these steps: 1) Evaluate the function at input values that are increasingly close to the point of interest, observing how the function's values change. 2) Analyze the function's graph, looking for the presence of vertical asymptotes, which indicate the existence of an infinite limit. 3) Combine the numerical and graphical information to conclude whether an infinite limit exists and, if so, determine its value (positive or negative infinity).
  • Explain the significance of identifying infinite limits in the context of understanding a function's behavior and properties.
    • Identifying infinite limits is crucial for understanding a function's behavior and properties, such as its domain and range. Infinite limits indicate points where the function's values approach positive or negative infinity, which can have important implications for the function's behavior. For example, the presence of a vertical asymptote due to an infinite limit means that the function is undefined at that point, and the function's domain must exclude that value. Additionally, understanding the locations and values of infinite limits can provide insights into the overall shape and characteristics of the function's graph, which is essential for analyzing and interpreting the function's behavior.