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In this section we introduce the concepts of limit and continuous function.

Let *f* (*x*) be a function from a subset of the real numbers to the real numbers, and let *x*_{0} be a
real number. We say that *f* (*x*) has a limit at *x* = *x*_{0} if as *x* approaches *x*_{0}, *f* (*x*) approaches
some number *L*. We call the number *L* the limit of the function *f* (*x*) at *x*_{0}, and we write

f (x) = L |

If *f* is the function whose graph is drawn below, then
*f* (*x*) = 2 and
*f* (*x*) = 5. Note that a function need not be defined at a particular value of *x*
(that is, *x* need not belong to the domain) in order for a limit to exist there.

Figure %: Limit Examples

It may be that a function has a limit at *x* = *x*_{0} only if *x* approaches *x*_{0} from one side
or the other. In fact, the function may have two different limits at *x*_{0}, depending on
the side from which *x* approaches *x*_{0}. Such limits are called one-sided limits. If *f*
has the limit *L* when *x* approaches *x*_{0} from the left, we say *f* has left-hand limit
*L* at *x*_{0} and write

f (x) = L |

We make the similar definition for the right-hand limit. An example of a function with one-sided
limits is shown below. What is the left-hand limit of *f* at -2? The right-hand limit?

Figure %: An Example of a One-Sided Limit

If a function *f* has a limit *L* at *x*_{0}, and *L* = *f* (*x*_{0}), then *f* is said to be
continuous at *x*_{0}. Note that this presupposes that *x*_{0} is in the domain of *f*, which
is not necessary when discussing limits in general. A function that is continuous at every
point in its domain is said to be a continuous function.

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