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2: Limits

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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 at a particular value of \(x\).

    • 2.0: Prelude to Limits
      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. 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: A Preview of Calculus
      As we embark on our study of calculus, we shall see how its development arose from common solutions to practical problems in areas such as engineering physics—like the space travel problem posed in the chapter opener. Two key problems led to the initial formulation of calculus: (1) the tangent problem, or how to determine the slope of a line tangent to a curve at a point; and (2) the area problem, or how to determine the area under a curve.
    • 2.2: 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.
    • 2.3: Infinite Limits
      We may also use limits to describe infinite behavior of a function at a point.  If a function has a one-sided limit that equals infinity at a point, the function will also have a vertical asymptote there.
    • 2.4: 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.
    • 2.5: Limits at Infinity
      In this section, we define limits at infinity and show how these limits affect the graph of a function.
    • 2.6: 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.
    • 2.7: 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.
    • 2R: Chapter Review Exercises

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

    Contributors

    Gilbert Strang (MIT) and Edwin “Jed” Herman (Harvey Mudd) with many contributing authors. This content by OpenStax is licensed with a CC-BY-SA-NC 4.0 license. Download for free at http://cnx.org.

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