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2: Limits

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  • Page ID
    213141
    • Gilbert Strang & Edwin “Jed” Herman
    • OpenStax

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    The idea of a limit is central to all of calculus. We begin this chapter by examining why limits are so important. Then, we go on to describe how to find the limit of a function at a given point. Not all functions have limits at all points, and we discuss what this means and how we can tell if a function does or does not have a limit at a particular value. This chapter has been created in an informal, intuitive fashion, but this is not always enough if we need to prove a mathematical statement involving limits. The last section of this chapter presents the more precise definition of a limit and shows how to prove whether a function has a limit.

    • 2.1: The Limit of a Function
      A table of values or graph may be used to estimate a limit. If the limit of a function at a point does not exist, it is still possible that the limits from the left and right at that point may exist. If the limits of a function from the left and right exist and are equal, then the limit of the function is that common value. We may use limits to describe infinite behavior of a function at a point.
    • 2.2: The Limit Laws
      In this section, we establish laws for calculating limits and learn how to apply these laws. In the Student Project at the end of this section, you have the opportunity to apply these limit laws to derive the formula for the area of a circle by adapting a method devised by the Greek mathematician Archimedes. We begin by restating two useful limit results from the previous section. These two results, together with the limit laws, serve as a foundation for calculating many limits.
    • 2.3: Limits at Infinity and Asymptotes
      To graph a function f defined on an unbounded domain, we need to know the behavior of f as x→±∞ . In this section, we define limits at infinity and show how these limits affect the graph of a function. At the end of this section, we outline a strategy for graphing an arbitrary function f.
    • 2.4: Continuity
      For a function to be continuous at a point, it must be defined at that point, its limit must exist at the point, and the value of the function at that point must equal the value of the limit at that point. Discontinuities may be classified as removable, jump, or infinite. A function is continuous over an open interval if it is continuous at every point in the interval. It is continuous over a closed interval if it is continuous at every point in its interior and is continuous at its endpoints.
    • 2.5: The Precise Definition of a Limit
      In this section, we convert this intuitive idea of a limit into a formal definition using precise mathematical language. The formal definition of a limit is quite possibly one of the most challenging definitions you will encounter early in your study of calculus; however, it is well worth any effort you make to reconcile it with your intuitive notion of a limit. Understanding this definition is the key that opens the door to a better understanding of calculus.
    • 2.R: Chapter 2 Review Exercises
      This page explores calculus concepts, focusing on limits and continuity. It examines true or false statements about function continuity and limit behavior, offering justifications and counterexamples. Exercises involve evaluating limits algebraically, identifying when limits do not exist, and using the squeeze theorem.

    Thumbnail: The function \(f(x)=1/(x−a)^n\) has infinite limits at \(a\). (CC BY; OpenStax)

    This page titled 2: Limits is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Gilbert Strang & Edwin “Jed” Herman (OpenStax) via source content that was edited to the style and standards of the LibreTexts platform.