Definition of a Limit

    The precise definition of a limit is given and it is shown through examples why the definition is needed. You may have already seen examples of estimating limits numerically and graphically. Each of these approaches produces an estimate of the limit, but it is the formal definition of the limit that allows us to prove results so that more analytic techniques for evaluating limits can be accomplished. In summary, a three-pronged approach to solving limits is often:

    (i) numerical approach by constructing tables of values
    
    (ii) graphical approach by sketching a graph by hand or using technology
    
    (iii) analytic approach by using algebra or calculus.

Definition (Limit of a Function) Suppose that the domain of f contains points x arbitrarily close to c but different from c. Then means, for all there exists , such that



for any in the domain of

    Concisely, a limit is used to describe the behavior of a function near a point but not at the point. The function need not even be defined at the point. If it is defined there, the value of the function at the point does not affect the limit. Intuitively, means we can make as close to as we wish by taking any sufficiently close to, but different from

Example (Necessity of a Formal Definition) We will use a guessing method to show why the formal definition of a limit is a necessity.

(a) Use a table to guess the values of



    Solution. From the table



The number is suggested to be Interestingly, if you try just to make sure you have taken numbers close enough to 0, you may find that the calculator gives the value 0. Does this mean that the limit is 0? No, the calculator may give you a false answer because when is small enough (like ) then seems like 0. But in fact is not equal to 0. The point is, using technology to verify a computation can lead to misunderstanding; and in fact, a formal definition of  a limit is needed. Using the formal definition of a limit, we can prove what the value of the limit is without any doubt. This type of proof is usually called an epsilon-delta proof since the formal definition is usually stated with the greek letters (epsilon) and (delta).

(b) Use tables of values to find the limit

    Solution. As before, we construct a table of values.



From the table it appears that   However, if we persevere with smaller values of the next table suggests





In fact, which is easily proven once the formal limit definition is used to prove some interesting limit rules and continuity is discussed.

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Cite this as:
Definition Of A Limit
Published by Library of Math -- Online math organized by subject into topics.
Written by Smith, David A.
http://www.libraryofmath.com/definition-of-a-limit.html