The Limit of a Function

    Intuitively speaking, the limit of a function is a real number that approaches as  x approaches a given real number c, this process is sometimes denoted by as and is usually denoted by   In order for a limit to exist there must be an infinite number of real numbers in the domain of f  that are arbitrarily close to c; but this alone is not sufficient for a limit to exist. Also it is important to note that the number c does not need to be in the domain of f. The main idea of the limit is to study the behavior of a function around a point in space.

Definition (Limit of a Function) Suppose that the domain of f contains points x arbitrarily close to c but different from c. Then



means that the functional values can be made arbitrarily close to a unique number by choosing sufficiently close to (but not equal to ).

    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

Proposition (Two-Sided Limits) The two-sided limit exists if and only if the one-sided limits and both exist and  



In which case,

Example (Finding a Limit by a Table) Find the limit of as approaches using a table of functional values for and

    Solution. We compute,  



Thus as x approaches from the left we estimate that approaches ; and as x approaches from the right we estimate that approaches Therefore, we estimate

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