MA101: Single-Variable Calculus I

2.3: Limits

• In this subunit, you will take a close look at a concept that you have used intuitively for several years: the limit. The limit asks the question, "what does the function do as the independent variable becomes closer and closer to a certain value?" In simpler terms, the limit is the natural tendency of a function. The limit is incredibly important due to its relationship to the derivative, the integral, and countless other key mathematical concepts. A strong understanding of the limit is essential to success in the field of mathematics.

• 2.3.1: The Definition and Properties of Limits

  • Whitman College: David Guichard's "Calculus, Chapter 2: Instantaneous Rate of Change: The Derivative, Section 2.3: Limits" URL

    Read Section 2.3 (pages 36-45). Read this section carefully, and pay close attention to the definition of the limit and the examples that follow. You should also closely examine the algebraic properties of limits as you will need to take advantage of these in the exercises.
    

  • Massachusetts Institute of Technology: David Jerison's "Limits" Page

    Watch this video.
    

  • Whitman College: David Guichard's "Calculus, Chapter 2: Instantaneous Rate of Change: The Derivative, Exercises 2.3, Problems 1 - 18" URL

    Work through problems 1-18 for Exercise 2.3. When you are done, check your answers against Appendix A.
    

  • University of California, Davis: Duane Kouba's "Precise Limits of Functions as x Approaches a Constant" URL

    Work through problems 1-10. When you are done, check your solutions against the answers provided.
    

• 2.3.2: The Squeeze Theorem

  • Whitman College: David Guichard's "Calculus, Chapter 4: Transcendental Functions, Section 4.3: A Hard Limit" URL

    Read section 4.3 (pages 75-77). The Squeeze Theorem is an important application of the limit concept and is useful in many limit computations. This reading teaches you a useful trick for calculating limits of functions where at first glance you might seem to be dividing by zero.
    

  • PatrickJMT: "The Squeeze Theorem for Limits" Page

    Watch this brief video. The creator of this video describes and illustrates the Squeeze Theorem by using specific examples.