Definition of a Limit

Read this section to learn how a limit is defined. Work through practice problems 1-6.

Definition of Limit

It may seem strange that we have been using and calculating the values of limits for a while without having a precise definition of limit, but the history of mathematics shows that many concepts, including limits, were successfully used before they were precisely defined or even fully understood. We have chosen to follow the historical sequence in this chapter and to emphasize the intuitive and graphical meaning of limit because most students find these ideas and calculations easier than the definition. Also, this intuitive and graphical understanding of limit was sufficient for the first hundred years of the development of calculus (from Newton and Leibniz in the late 1600's to Cauchy in the early 1800's), and it is sufficient for using and understanding the results in beginning calculus. 

Mathematics, however, is more than a collection of useful tools, and part of its power and beauty comes from the fact that in mathematics terms are precisely defined and results are rigorously proved. Mathematical tastes (what is mathematically beautiful, interesting, useful) change over time, but because of these careful definitions and proofs, the results remain true, everywhere, and forever. Textbooks seldom give all of the definitions and proofs, but it is important to mathematics that such definitions and proofs exist. 

The goal of this section is to provide a precise definition of the limit of a function. The definition will not help you calculate the values of limits, but it provides a precise statement of what a limit is. The definition of limit is then used to verify the limits of some functions, and some general results are proved. 

Source: Dave Hoffman,
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